Equivalence Definition and 721 Threads

According to https://plato.stanford.edu/entries/zermelo-set-theory/ , Zermelo (translated) said: I don't know if that quote is part of his formal presentation. It does raise the question of whether set theory must formally assume that there exists an equivalence relation on "elements" of...

(a) I present the following counter example for this. Let ##A = \{0,1,2,\ldots \}## and ##B = \{ 2,4,6, \ldots \} ##. Also, let ##C = \{ 1, 2 \} ## and ##D = \{3 \}##. Now, we can form a bijection ##f: A \longrightarrow B## between ##A## and ##B## as follows. If ##f(x) = 2x + 2##, we can see...

Hi all, I have ran into some confusion about the equivalence principle; perhaps I should state what I understand and then proceed to ask questions. It is my understanding that the equivalence principle states that spacetimes are locally Minkowski, and so the rules of SR apply in that locality...

Hi equivalent class is a Cartesian product of A*A. Then shouldn't it's union be a partition of A*A, instead being a partition of A

Hey! :o Let $M:=\{1, 2, \ldots, 10\}$ and $\mathcal{P}:=\{\{1,3,4\}, \{2,8\}, \{7\}, \{5, 6, 9, 10\}\}$. For $x \in M$ let $[x]$ be the unique set of $\mathcal{P}$ that contains $x$. We define the relation on $M$ as $x\sim y:\iff [x]=[y]$. Show that $\sim$ is an equivalence relation. For...

Hey! :o Consider the following metrices in $\mathbb{R}^2$. For $x,y\in \mathbb{R}^2$ let \begin{align*}&d_1(x,y)=|x_1-y_1|+|x_2-y_2| \\ &d_2(x,y)=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2} \\ &d_{\infty}(x,y)=\max \{|x_1-y_1|,|x_2-y_2|\}\end{align*} Draw the unit ball $B_i(0,1)=\{y\in X\mid d_i(0,y)<1\}$...

According to wikipedia, the strong equivalence principle states “the gravitational motion of a small test body depends only on its initial position in space time and velocity, and not on its constitution, and the outcome of any local experiment (gravitational or not) in a freely falling...

I'm an amateur physics enthusiast, and there is a question that's been in the back of my mind for some time that I haven't been able to answer on my own, and haven't gotten a satisfactory answer elsewhere. First, I want to define a couple of terms and make sure my understanding isn't breaking...

Hi, today I stumbled upon a 2016 article in Scientific American about the (then) possibility of re-defining the kilogram through Planck's constant. The article is really a very quick review of the topic. At some point the author states the following "So for years, physicists have chased an...

Hello World, I have understood the following: in SR, time intervals and space intervals (distances, lengths) are relative and inertial reference frame dependent. Space and time is not absolute anymore. However, acceleration is still absolute: different inertial frames see the same acceleration...

I have a 4D array of dimension ##100\text{x}100\text{x}3\text{x}3##. I am working with `Python Numpy. This 4D array is used since I want to manipulate 2D array of dimensions ##100\text{x}100## for the following equation (it allows to compute the ##(i,j)## element ##F_{ij}## of Fisher matrix) ...

Consider the equivalence: (∀v Fv -> p) <=> (∃u Fu -> p) Where variable v occurs free in Fv at all and only those places that u occurs free in Fu, and p is a proposition containing no free occurences of variable v. Can someone please offer a proof of such equivalence. Many thanks. am

Equations of attraction or repulsion can get very complicated when the field shapes and densities are not identical. Intuition would hold the forces to be the same, but in on closer examination the field shape of two repulsing magnets looks entirely different from two attracting magnets. I...

Can someone please explain what the tilde represents? We have had no info on this to date. I know it has to do with an equivalence relation but not sure what it represents on its own as in part (a) of the attached. Just want to make sure I'm clear. Thanks!

Hi Lets start off with the definition of diffeomorphism from Wolfram MathWorld: The issue is that I am learning about smooth manifolds, and in the books I've read, the map has to be smooth and have a smooth inverse. Also, the definition above doesn't say that it has to be bijective. However...

Let ##X = \{x_1 , \dots , x_n\}##. Then ##\text{aff}(X) = \text{intersection of all affine spaces containing X}##. Let ##C(X)## be the set of all affine combinations of elements of ##X##. We want to show that these two sets are equal. First we focus on the ##\text{aff}(X) \subseteq C(X)##...

Greetings, I am searching for an explanation that connects the mean squared displacement in an ensemble of particles to the velocity autocorrelation functions. I know that they are connected since both of them can be used to calculate the diffusion constant. But I do not find a mathematical...

I am currently studying Fisher's formalism as part of parameter estimation. From this documentation : They that Fisher matrix is the inverse matrix of the covariance matrix. Initially, one builds a matrix "full" that takes into account all the parameters. 1) Projection : We can then do...

Consider an electric dipole consisting of charges ##q## and ##-q##, both of mass ##m##, separated by a distance ##d##. If the dipole is given an acceleration ##a## perpendicular to its moment the total electric force on it, due to each charge acting on the other, is given approximately by...

Dear Everyone, $\newcommand{\R}{\mathbb{R}}$ I am struck in writing the equivalence classes. And the problem is this: Let ${\R}^{2}= \R \times \R$. Consider the relation $\sim$ on ${\R}^{2}$ that is given by $({x}_{1},{y}_{1}) \sim ({x}_{2},{y}_{2})$ whenever...

Let ##d_1## and ##d_2## be two metrics on the same set ##X##. We say that ##d_1## and ##d_2## are equivalent if the identity map from ##(X,d_1)## to ##(X,d_2)## and its inverse are continuous. We say that they’re uniformly equivalent if the identity map and its inverse are uniformly...

I understand that the first part of the equation is an equivalence class due to reflexivity, symmetry, and transivity... but I am confused on the second part. Could someone please help me out? THANKS

I've seen a proof that the path integral formulation of quantum mechanics is equivalent to solving Schrodinger's equation. However, it appears to me that the proof actually depended on the Hamiltonian having a particular form. I'm wondering how general is the equivalence. Let me sketch a...

I am working on a set equivalent (the textbook refers as "equinumerous" denoted by ~) as follows: If $S$ and $T$ are sets, prove that if $(S\backslash T) \sim (T\backslash S)$, then $S \sim T$. Here is my own proof, I am posting it here wanting to know if it is valid. (It may not be as elegant...

Einstein's equivalence principle states that: The sets of inertial frames in the real world that correspond to (portions of) the ideal set of inertial frames discussed in special relativity consist of freely falling local frames. In other words,can we say that since all the local frames are in...

In Newtonian mechanics, both gravitational mass and inertial mass is m. This principle is known as the principle of equivalence. However, I heard that in Relativity, gravitational mass is γm instead of m because total energy of the particle is γmc2. But in special relativity, it is widely known...

Hey! :o Let $\mathbb{K}$ be a field, $V,W$ finite dimensional $\mathbb{K}$-vector spaces and $\Phi:V\rightarrow W$ a linear map. I want to show that the following two propositions are equivalent: $\Phi$ is surjective For each linear form $\beta:W\rightarrow \mathbb{K}$ it holds...

I am newbie to topology and trying to understand covering maps and quotient maps. At first sight it seems the two are closely related. For example SO(3) is double covered by SU(2) and is also the quotient SU(2)/ℤ2 so the 2 maps appear to be equivalent. Likewise, for ℝ and S1. However, I...

<Moderator's note: Moved from a technical forum and thus no template.> Not sure this should be under Linear and Abstract Algebra, but regardless I need help with a question in my mathematical proofs course. Here it is: Let ∼ be a relation defined on Z by x ∼ y if and only if 5 | (2x + 3y). (a)...

A recently published paper discusses the necessity of applying Einstein’s Equivalence principle to the quantum realm. It’s validity adds mor constraints on how quantum particles may move. https://phys.org/news/2018-08-einstein-equivalence-principle-quantum-world.html

We have a cube on an inclined plane. The tipping condition is the presence of an unbalanced torque relative to the center of mass (contributing forces are: the normal force and the force of friction). However, is this conditions equivalent to the previous one: The line of action of the force of...

Does the mass-energy equivalence mean that if we heat a body then its mass will increase?

What is the difference between 'equivalent and equivalence'?

If we are representing the basis vectors as partial derivatives, then ##\frac{\partial}{\partial x^\nu + \Delta x^\nu} = \frac{\partial}{\partial x^\nu} + \Gamma^\sigma{}_{\mu \nu} \Delta x^\mu \frac{\partial}{\partial x^\sigma}## to first order in ##\Delta x##, correct? But in the same manner...

Homework Statement 1) for a given reaction to consume one reactant completely, must the equivalents of both reactants be same? for example, I know in the reaction of HCl + NaOH - the equivalents of HCl=equivalents of NaOH for a titration, is it the same for Na2CO3 + HCl? 2) the following is an...

Consider 2 lifts ,one on ground and other on acceleration, principle of equivalance says you can't find you are on a gravitational field, or accelerating. g decrease when r increase so I can find which lift on gravitational field ?

If it takes anywhere between 5,000 to a couple of hundred thousand years (various internet sources have various values) for photons generated in the sun’s core to reach its surface and radiate out, what is the estimated mass equivalence of all these photons making their way out from the core to...

Is there a controversy over the EP? What I mean is: is it considered to be false beyond any doubts or on the contrary it is absolutely true and doubts about its validity are only misinterpretations? I know that the question it's not so simple because there are many EPs, strong, weak, etc. but I...

10.) neg[r\implies neg(w v s)] 13.) r\implies neg(w v s) ----------------------------- R R^(wvs) --------------------------...

Hello all, I was doing some behavioural modelling of the torque transfer characteristics of a belt drive system from the driver pulley to the driven pulley. While doing the same, i have tried to see how the angular velocity is getting transferred as well. I would explain my point with the...

[Moderator's note: Spun off from another thread due to topic change.] Can I say that "being at rest in a uniform gravitational field is locally equivalent to accelerating in flat spacetime, free falling in a uniform gravitational field is locally equivalent to nonacceleration in flat spacetime ?"

Some time ago there was a similar thread https://www.physicsforums.com/threads/clock-hypothesis-gravity-time-dilation-and-equivalence-principle.929838/ but what I want to discuss is similar but not the same and I would like to specify my question in such way, that it hopefully won't go sideways...

I think mass as a form of potential energy and am always told that this is wrong. According to wiki: "In physics, potential energy is the energy possessed by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors." Why do this...

Equivalence principles explains why lite bend towards massive objects, Einstein uses a moving lift to illustrate this, the light will seem to be bending if the lift is moving, but for a stationary lift, it will not because the position it strikes is stationary. So I think it is not correct...

Take the following statement as given: ##\forall \epsilon > 0 ~ (|x-y| < \epsilon) \implies x = y##. To convert this prenex normal form, we get ##\exists \epsilon > 0 ~ (|x-y| < \epsilon \implies x = y)##. How are these two statements equivalent? It seems as though they are saying different...

In Principle of equivalence, we indroduce to the theoram by a lift experiment, my question is why the lift is fully closed one, why the observer in lift forbidden to observe out side world

Homework Statement Part C ) help Homework Equations So far.. Moment resultant = 1800i +3200k , magnitude = 3070 N*m Force resultant = 500i +300j + 800k , magnitude = 990 N The Attempt at a Solution For the location, I'm not getting x = 1.16, y =2.06. Instead , I am getting Mx = Fr*y --> y...

<< Mentor Note -- Thread moved from the technical forums, so no Homework Template is shown >> Okay, so i am having severe problems with figuring out what i did wrong... I am given : F1 = 250 and F2 = 90 The correct Force result is = 245 i +228 j with magnitude of 335 . The moment given is...

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of Corollary 1.8.10 ... ... Duistermaat and Kolk's Corollary 1.8.10 and the preceding notes and results read...

So , what I was wondering about was a slight difference in notation, for which I am not certain if correct (mine, in particular.). The induction axiom says: If M is a subset of ℕ, and if holds that: a)1∈M b)(∨n∈ℕ)(n∈M→s(n)∈M) then M=ℕ. Now my question is: why do we write (∨n∈ℕ)(n∈M→s(n)∈M)...