Space Definition and 1000 Threads

Suppose we have an infinite dimensional Hilbert-like space but that is incomplete, such as if a subspace isomorphic to ##\mathbb{R}## had countably many discontinuities and we extended it to an isomorphism of ##\mathbb{R}^{\infty}##. Is there a measure of integrating along any closed subset of...

People under 20 have never lived at a time without at least two people in space. Exactly 20 years ago, on Halloween 2000, Soyuz TM-31 launched the first long-term crew towards the ISS to start Expedition 1. They docked 2 November. Since then the ISS has been inhabited continuously. This was by...

Hi. I don't know what prefix this question belongs in so I just chose advanced at random. What's the physical effect called when the Earth orbits around the sun at extremely fast speeds and also rotates around itself every 24 hours at the same time? Does that force cause anything in space...

The question I am trying to solve is what is the velocity vector (direction and magnitude) of an object in 2 d space. We know the distance measured to the car from two different angles. We know the radial velocity of the car on both measurements. The radial velocity is the component of the...

The hamiltonian ´for a free falling body is $$H = \dfrac{p^2}{2m} + mgy$$ and since we are using cartesian coordinates that do not depend on time and the potential only depends on the position, we know that ##H=E##. For this hamiltonian, using the Hamilton's equations and initial conditions...

Attached is the schematic for the circuit. It uses a TPS61042DRBT LED driver along with a PSoC 4000 8pin microcontroller to drive a 10mA LED with push button controls for brightness. The problem is some components, like the inductor and sense resistor is way too large (over 6mm long!). This is...

I have a question about building efficient heat engines in outer space. In theory you could have a hot reservoir heated by the sun that was several hundred degrees C, and a cold reservoir that was very cold - maybe 50K - 100K or even colder. Thus, theoretically at least, a heat engine could be...

Is there any relation between dynamical system and Hilbert space(functional analysis)?

Consider two 1 square meter marble slabs each of mass 1 kg floating in space facing the sun such that light from the sun incident perpendicularly on the flat faces. At equilibrium, power received from the sun 'PS' equals the power being lost in the form of radiation 'PR'. PS = PR (at...

Build a tube to space. Use our atmosphere to push a ship up the tube.

Be f a orthogonal transformation and g being in the canonical form. $$[f^{-1}]^{t}[g][f^{-1}] = [g']$$ So this equation of isometries implies that the diagonal of g is +- 1, but, apparently, if it is minus one it can't be a group, and i don't know why. To make the things clear, if det = +1, in...

The answer to the primary question in the summary is the first step in seeking an answer to a more complicated question I plan to post in a separate thread later. This more complicated question is a consequence of the thread...

I assume 3-D cartesian space is not an adequate description for things at the cosmological scale. So what definition of "space" is used when people talk about things like the distribution of hydrogen atoms "in the universe"?

Is space travel a possibility?

"The dual space is the space of all linear maps from the original vector space to the real numbers." Spacetime and Geometry by Carroll. Dual space can be anything that maps a vector space (including matrix and all other vector spaces) to real numbers. So why do we picked only a vector as a...

If I'm given a set of four vectors, such as A={(0,1,4,2),(1,0,0,1)...} and am given another set B, whose vectors are given as a form such as (x, y, z, x+y-z) all in ℝ, what steps are needed to show A is a basis of B? I have calculated another basis of B, and found I can use linear combinations...

Let V = C[x] be the vector space of all polynomials in x with complex coefficients and let ##W = \{p(x) ∈ V: p (1) = p (−1) = 0\}##. Determine a basis for V/W The solution of this problem that i found did the following: Why do they choose the basis to be {1+W, x + W} at the end? I mean since...

Let ## \mathcal{S} ## be a family of probability distributions ## \mathcal{P} ## of random variable ## \beta ## which is smoothly parametrized by a finite number of real parameters, i.e., ## \mathcal{S}=\left\{\mathcal{P}_{\theta}=w(\beta;\theta);\theta \in \mathbb{R}^{n}...

I draw the graph like this: For (b), I divided each force vector to e from p1 and p2 as x and y parts. I computed them and got Fx=-4.608*10^(-15)N Fy=-2.52*10^(-15)N However, I am not sure whether I did it correctly or not... I appreciate every help from all of you! Thank you!

Lets go through the example problem until we get to the part I don't understand. Figure 25-17 can be used as a reference to all questions. From part (a) to part (b) we eventually find the charge q on one plate (and by default the charge -q on the other). No problem there. The battery is then...

Sorry if the question is not rigorously stated.Statement: Let ##(q,p)## be a set of local coordinates in 2-dimensional symplectic space. Let ##\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})## be a set of local coordinates of certain open set of a differentiable manifold ##\mathcal{M}.## For...

I would like to show that fixing the orientation of k-manifold smooth connected ##S## in ##\mathbb {R} ^ n ## is equivalent to fixing a frame for one of its tangent spaces. What I know is that different orientations correspond to orienting atlases containing maps that cannot be consistent with...

Rookie question; for a vector space ##V##, with basis ##v_1, v_2, \dots, v_n##, the dual space ##V^*## is the set of linear functionals ##\varphi: V \rightarrow \mathbb{R}##. Dual basis will satisfy ##\varphi^i(v_j) = \delta_{ij}##. Is the action of any dual vector on any vector always an inner...

hi i was recently introduced to the Dirac notation and i guess i am following it really well , but can't get my head around the idea that the bra vector said to live in the dual space of the ket vectors , i know about linear transformation and the structure of the vector spaces , and i realize...

In Dynamical Systems Theory, a point in phase space is interpreted as the state of some system and the system does not exist in two states simultaneously. Can some phase spaces be given an additional interpretation as describing a field of values at different locations that exist...

I was recently working on the two body problem and what I can say about solutions without solving the differential equation. There I came across a problem: Lets consider the Kepler problem (the two body problem with potential ~1/r^2). If I use lagrangian mechanics, I get two differential...

If Evangelista Torricelli truly created a vacuum, then there would be nothing in it, yet you can see through it which means light is obviously still in there (and who knows what else), right? If there was truly nothing in it, and glass is a highly viscous fluid, and fluids conform to fill empty...

All help is welcome!

Is it possible to create following two shapes from glass using currently known glass mass production techniques? Shape #1: bottle with a prolonged neck that continues into the inner space, like this (cross section): | | | | /| |\ / | | \ / \ / \ | | |...

At the core of the earth, or sun, there is no net force of gravity...every direction is up...does this mean that space is not curved, or more generally, becomes less curved from the surface of an object towards its center?

I have heard of phase space path integrals, but couldn't find anything in Wikipedia about it, so I am wondering, what does it compute ? In particular, are the endpoints points of definite position and momentum? If so, how does one convert them to quantum states ? Also, how is it related to...

Hello everybody, my question may sound stupid, especially speaking of such a mind-blowing and important theory... but here I am! I'm 17 and I'm reading a fabulous book by Stephen Hawking, "A Brief History of Time", and it introduced me to relativity theories... I literally started looking the...

If one were trying to look for life outside of our planet, would it be more logical to start off on Mars or to go elsewhere?

Can time and space be thought of fields? And if so, how can this affect spacetime? And then what would this mean for affect general relativity?

Does the concept of the angle between two vectors make sense in Minkowski space? Does the concept of orthogonal basis for Minkowski space make sense? If it does, how is it defined? When we start with the usual (time, distance) basis for 2-D Minkowski space, the axes as drawn make a right...

Did energy begin to exist at the Big Bang? Can energy exist without space and time? Or don't we know? When I've tried to research this I get a mix of different answers. I have virtually no understanding of science or physics in general FYI.

Where do I start. I want to write the matrix form of a single or two qubit gate in the tensor product vector space of a many qubit system. Ill outline a simple example: Both qubits, ##q_0## and ##q_1## start in the ground state, ##|0 \rangle =\begin{pmatrix}1 \\ 0 \end{pmatrix}##. Then we...

[I urge the viewer to read the full post before trying to reply] I'm watching Schuller's lectures on gravitation on youtube. It's mentioned that spacetime is modeled as a topological manifold (with a bunch of additional structure that's not relevant to this question). A topological manifold is...

Hello As you know, the geometric definition of the dot product of two vectors is the product of their norms, and the cosine of the angle between them. (The algebraic one makes it the sum of the product of the components in Cartesian coordinates.) I have often read that this holds for Euclidean...

If the spin space is independent of the real space, what is the meaning of, for example, the z-component of the spin?

Hi I believe I understand the concept of a vector space V and its dual V*. I also understand that for V finite dimensional, there is a natural isomorphism between V and V**. What I am struggling to understand is - Does this natural isomorphism mean that V** is always IDENTICAL to V (identical...

This is section 16.3 of QFT for the Gifted Amateur. I understand the concept of the spacetime propagator ##G^+(x, t, x', t')##, but the following propagator is introduced without any explanation I can see: $$G^+(x, y, E) = \sum_n \frac{i\phi_n(x)\phi_n^*(y)}{E - E_n}$$ It would be good to have...

I'm studying 'A Most Incomprehensible Thing - Notes towards a very gentle introduction to the mathematics of relativity' by Collier, specifically the section 'More detail - contravariant vectors'. To give some background, I'm aware that basis vectors in tangent space are given by...

Thank you all for clearing my doubt before. There is a question I want to ask on space and time and this time it is not about the absolute time as I have understood fairly that space and time are always relative. So Here it is according to GR It is very beautifully explained that Very massive...

Suppose I'm traveling inside a spaceship at speed comparable to light between two points A and B. According to me the distance between the two points will be shortened due to length contraction. But actually my spaceship passes through every point between A and B so the distance measured by...

Good Morning Recently, I asked why there must be two possible solutions to a second order differential equation. I was very happy with the discussion and learned a lot -- thank you. In it, someone wrote: " It is a theorem in mathematics that the set of all functions that are solutions of a...

I'm studying 'Core Principles of Special and General Relativity' by Luscombe - the chapter on tensors. Quoting: The book goes on to talk about a switch to the spherical coordinate system, in which ##\mathbf{r}## is specified as: $$\mathbf{r}=r\sin\theta\cos\phi\ \mathbf{\hat...

Summary:: Inner Product Spaces, Orthogonality. Hi there, This my first thread on this forum :) I encountered the above problem in Schaum’s Outlines of Linear Algebra 6th Ed (2017, McGraw-Hill) Chapter 7 - Inner Product Spaces, Orthogonality. Using some particular values for u and v, I...

This is an experiment I would have liked to do from the ISS, but an approximation could be done in a vacuum chamber on Earth. How big a soap bubble, polymer bubble, or glass bubble could you blow in the vacuum of space? How to calculate the evaporation rate in vacuum? A liquid exposed to vacuum...

Imagine this question in 2 dimensions, time (t) and distance (x), that is (t,x). Alice (A) is at the origin, x=0. Bob (B) begins at x=c. Thus we have A(0,0) and B(0,c). Both Alice and Bob send a light signal towards the other but let's say the signal changes colour every second by the colours of...