Local Definition and 491 Threads

Spin (and therefore photon circular polarization) can be constructed generically in terms of intrinsic rotations on vector fields, whether describing classical and quantum physical systems: e.g. https://scholar.google.co.uk/scholar?cluster=14889979702374754652&hl=en&as_sdt=0,5 Building on this...

Alan Macdonald claims in "4 Appendix: The Equivalence Principle" of his text "Special and General Relativity based on the Physical Meaning of the Spacetime Interval", that his calculation regarding a 2D-surface of a sphere proves, that the equivalence principle is violated. He defines the...

These papers claim to present a realistic stochastic interpretation of quantum mechanics that obeys a stochastic form of local causality. (A lecture I recently attended mentioned these papers). It also claims the Born rule as a natural consequence rather than an assumption. This appears to me to...

Let ##f## be a measurable function supported on some ball ##B = B(x,\rho)\subset \mathbb{R}^n##. Show that if ##f \cdot \log(2 + |f|) ## is integrable over ##B##, then the same is true for the Hardy-Littlewood maximal function ##Mf : y \mapsto \sup_{0 < r < \infty}|B(y,r)|^{-1} \int_{B(y,r)}...

Pseudo-Riemannian manifolds (such as spacetime) are locally Minkowskian and this is very important for relativity since even in a highly curved spacetime, one could locally approximate the spacetime into a flat minkowski one. However, this would be an approximation. Perhaps this is a naive...

Let ##X## be a topological space, and let ##\mathscr{F}## be a sheaf on ##X##. Show that if ##\mathscr{U}## is an open cover of ##X## such that the restriction ##\mathscr{F}|_U## is flasque for every ##U\in \mathscr{U}##, then ##\mathscr{F}## is flasque. Note: A sheaf ##\mathscr{G}## on ##X##...

Are there any thoughts on what effects a Kimberlite eruption would have on nearby populations, if one occurred in modern history? They were supposedly quite violent based on the estimated rates of ascent of the diamond-bearing lava, but how does that translate to VEI or megatons, etc. ? [...

We do not seem to have any unexplained orbital/gravitational anomalies within the solar system. What does that imply for the local dark matter distribution?

Let ##f : X \to Y## be a surjective continuous closed map of topological spaces such that every fiber ##f^{-1}(y)## is compact. Show that ##Y## is locally compact if ##X## is locally compact.

Quantum Bayesianism takes the view that the there are no quantum states in the objective sense and that the probabilities should only be interpreted as what information an agent has about the system. Isn't this the same as claiming that there are hidden variables, and that probabilities arises...

I found an old article (https://journals.aps.org/pr/abstract/10.1103/PhysRev.137.B1379) which talks about conservation of energy in an expanding space. Apparently, the author found that energy is conserved at local scales (like the motion of planets in our solar system) as one would expect, but...

This question was raised but not answered in a thread which is now permanently closed. Consider the local conservation of charge ##\partial_{\mu}j^{\mu}=0##. In quantum field theory it is valid as an operator identity, but operators as such do not have a direct operational (experimental)...

Hi, Also, I read this article, What Do You Mean, The Universe Is Flat? (Part I), on Scientific American; URL: https://blogs.scientificamerican.com/degrees-of-freedom/httpblogsscientificamericancomdegrees-of-freedom20110725what-do-you-mean-the-universe-is-flat-part-i/ I have few questions about...

How does the Milky Way galaxy move in the local Group? Is there a circular motion around the center of the local Group like the sun moves around the center of the galaxy?

Given that I have a little personal app, developed on pc, tested (under iOS) and running. How can I get the app on my iOS device? All I have is a folder with a lot of files in it.

Spacetime is a differential manifold and at each point is attached a Minkowski spacetime. There the laws of physics are the usual ones without gravity. Gravity is the curvature of spacetime. To define the concept of curvature do we need to evaluate at least one neighborhood of point P? Is...

I am stuck deriving the gauge field produced in curved spacetime for a complex scalar field. If the underlying spacetime changes, I would assume it would change the normal Lagrangian and the gauge field in the same way, so at first guess I would say the gauge field remains unchanged. If there...

I have information that $$\rho_{ab}=\sum_{j}p_{j}\ket{\Psi_{j}^{ab}}\bra{\Psi_{j}^{ab}}$$ and $$Pr(o_{j}^{(a)}|\Psi_{ab})=Tr_{ab}(\ket{\Psi_{ab}}\bra{\Psi_{ab}}(\ket{o_{j}^{(a)}}\bra{o_{j}^{(a)}}\otimes \mathbb{I}_{2})) \text{.}$$ I started by representing the density operator for pure states...

The EM field seems to be required for for local gauge symmetry of the electron matter field under local phase variation. Following is a description (not my verbiage): There is a symmetry in physics which we might call the Local Phase Symmetry in quantum mechanics. In this symmetry we change...

This code is for iterative deepening depth first search in python. # Python dictionary to act as an adjacency list graph = { '7' : ['19','21', '14'], '19': ['1', '12', '31'], '21': [], '14': ['23', '6'], '1' : [], '12': [], '31': [], '23': [], '6' : [] } visited=[] goal='31'...

This code is for iterative deepening depth first search in python. # Python dictionary to act as an adjacency list graph = { '7' : ['19','21', '14'], '19': ['1', '12', '31'], '21': [], '14': ['23', '6'], '1' : [], '12': [], '31': [], '23': [], '6' : [] } visited=[] goal='31'...

Gauge symmetry is highly confusing, partly because many definitions differ in the literature. Strictly speaking gauge symmetry should be called gauge redundancy since you are mapping multiple representations to the same physical state. What is your favourite definition of what "large" gauge...

Hi, I'm reading Zavialov's book on QFT and there's a statement there I was interested in finding how to prove it. The statement is as follows: If ##[A(x), j_{\{\lambda\}}(y)]=0## for ##(x-y)^2<0## then ##A(x)## is a local polynomial. The relevant definitions are: $$A = \sum_n \int...

Consider the following theorem: Theorem: Let ##f## be a concave differentiable function and let ##g## be a concave function. Then: ##y \in argmax_{x} {f(x)+g(x)}## if and only if ##y \in argmax_{x} {f(y)+f'(y)(x-y)+g(x)}.## The intuition is that local maxima and global maxima coincide for...

Hey ! :giggle: Let $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$, $f(x,y)=\sin^2(x)\cdot \cos^2(y)$. - Show that $f$ has at $\left (\frac{\pi}{2}, 0\right )$ a strictly local maximum and that is also a global maximum. - Determine all points at which $f$ gets its global minimum. I have sone...

Does the position of the origin for the body’s rotating coordinate frame 1) stay fixed to the moving body or 2) does it stay fixed to the inertial frame, yet still able to rotate as the body rotates with the only restriction that it cannot translate with the body i.e. only affixed at the...

I am not sure where I should be posting this; I am perfectly aware this is Taylor expansion calculus, but considering the fact that I am working with numerical ODE solutions, I posted this here. In the paper: Implementation of an Adaptive BDF2 Formula and Comparison with the MATLAB Ode15s...

Hey! :giggle: We have the function $\displaystyle{f(x,y)=y^2-3x^2y+2x^4}$ and the function $\displaystyle{g_v(t)=f(tv_1, tv_2)=t^2v_2^2-3t^3v_1^2v_2+2t^4v_1^4}$. I have shown that $g$ has a local minimumat $t=0$ I want to show that $f$ has not a local minimum in $(0,0)$. The gradient is...

Because the local red fox population is feeling amorous tonight. And do you know what amorous red foxes sound like? If I use any descriptors that come to the forefront of my mind, Mr. Berkeman is likely to give me another weinie point for offending somebody. You're going to have to listen to...

Newton thought of gravity as action at a distance. Einstein showed that gravity is the curvature of space-time. Einstein's General Relativity is the best answer, but neither discusses gravity at the quantum level. I tend to think of gravitational force to be the result of matter interacting with...

I've come to a grinding halt with this and I can't see a way forward. Can someone please take a look at what I've done so far and let me know if what I have done is OK and then if it is, give me a hint on how to proceed. First up, Is ## u \cdot \nabla \cdot T = u_\alpha...

The ODE given to us is y' = xcosy. I am having a bit of trouble when it comes to solving this problem. We are supposed to show that the solution has a local minimum at x = 0 with the hint to think of the first derivative test. However, I am only really familiar with the first derivative test...

Just read this paper: https://arxiv.org/pdf/1906.04313.pdf At first it had me thinking that locally mediated, future dependent interpretations are the way to go. Yet it admits these seem to be rare relative to other types of interpretations. Any good intuition or reasons why this is rare in...

Under the Schrodinger Picture, nonrelativistic Quantum Mechanics for a fixed number of particles is highly nonlocal, e.g. Quantum Entanglement. But Quantum Field Theory is local. Why is that? Is it because QFT was created to accommodate SR, which, as a classical theory, is local? As always...

Find where increasing/decreasing, concavity, local extrema and inflection points for f(x)=ln/x So here is what I have so far: The derivative is 1-ln(x)/x^2 Critical points are (e,1/e) No concavity Local max is also (e,1/e) (no local min) no inflection points Increase on (0, e) and...

Hello. The recent discovery of the galaxy SPT0418-47 has piqued my interest. https://www.almaobservatory.org/en/images/reconstructed-view-of-spt0418-47/ https://en.wikipedia.org/wiki/SPT0418-47 https://www.eso.org/public/archives/releases/sciencepapers/eso2013/eso2013a.pdf It's my current...

I have a local repository (let's call it MyRepo) that is linked with Github (origin). The master branch is set to track origin/master using $ git branch -u origin/master Now I create another branch in my local repo, add a file in that branch, commit the change and merge the branch with master...

I'm trying to find the local truncation error of the autonomous ODE: fx/ft = f(x). I know that the error is |x(t1) − x1|, but I can't successfully figure out the Taylor expansion to get to the answer, which I believe is O(h^3). Any help would be greatly appreciated!

From Zorich, Mathematical Analysis II, 1st ed., pag.163: where the referred mapping (12.1) is a map ##\varphi:I_k\to U_S(x_0)##, in which: 1. ##I_k\subset\mathbb{R}^n## is the k-dimensional unit cube, 2. ##x_0## is a generic point on the surface ##S## and ##U_S(x_0)## is a neighborhood of...

Okay, so my algorithm looks something like this: ==== 1. Locate mid-point of the interval . This is our estimate of the local max. 2. Evaluate . 3. Divide the main interval into two subintervals: a left and right of equal length. 4. Check to see if there is a point in the interval such that ...

So this is a very long answer for a question worth 4 marks. Would it be a good idea to keep the explanation, or to have it strictly math only?

Fred H. Croom (Principles of Topology) and Tej Bahadur Singh (Elements of Topology) define local basis (apparently) slightly differently ... Croom's definition reads as follows:... and Singh's definition reads as follows: The two definitions appear different ... ... Croom requires that each...

Some students I mentor in the SE US have reported that their local PGRE has been cancelled, but additional info is hard to come by. They are trying to find out if it is still being offered on that date at other locations. What do y'all know? Anyone know of a location in the US where it is...

Temperature is often imagined as a scalar field varying over (physical) space. Yet temperature seems to be generally defined as the (reciprocal) rate of change (with fixed V,N) of the entropy of a system w.r.t. the energy of that system; a seemingly global property. Is this definition only for...

If you are floating in space in your spaceship and you kick in the engines and accelerate at a comfortable 1G and you end up standing on the bottom of your ship, a slight curvature of space-time is formed, throughout the ship, perhaps immeasurable, such that without windows on the ship, you...

Hi all, Let me give some background to my question. In computational neuroscience it's now fashionable to train recurrent neural networks (RNNs) to solve some task (working memory for example). We do so by defining some cost function and training the model weights to minimize it (using...

Hi PF! I have data that I need to interpolate (don't want to go into details, but I HAVE to interpolate it). I'm trying to find the local maximas on a given domain. I've looked everywhere and still haven't been able to do it? Seems most people work with NDSolve, but I don't use that function...

Hi All I am a bit exasperated right now. On another forum a person claimed Bell's second theorem proved QM was not local. I carefully explained what local causality was, and what the theorem states: There exist quantum phenomena for which there is no theory satisfying local causality. It...

First some background, then the actual question... Background: (a) Very simple example: if we take ##Asin(x+ϕ)+0.1##, the average is obviously 0.1, which we can express as the integral over one period of the sine function. (assume that we know the period, but don't know the phase or other...