Equivalence Definition and 721 Threads

Homework Statement Taking e to be small, we have been given these two formulas to prove m1 = \frac{-1+\sqrt{1-4e}}{2e} = \frac{-1+(1-2e+O(e^2))}{2e} = -1+ O(e) m2= \frac{-1-\sqrt{1-4e}}{2e} = \frac{-1-(1-2e+O(e^2))}{2e} = -1+ O(1) Homework Equations The Attempt at a Solution...

It is my understanding that the Equivalence Principle postulates that if I were standing in a closed room, I would not be able to distinguish whether the downward force that I felt was caused by (1) the presence of a massive body such as the Earth exerting a downward gravitational force or (2)...

Let A be a set. For every set B and total function f:A->B we define a relation R on A by R={(x,y) belonging to A*A:f(x)=f(y)} *belonging to - because i don't know how to make the symbole... Prove that f is one-to-one if and only if the equivalence classes of R are all singletones

I've been trying to put this together in my head for a while and it doesn't come out. For example, if there are more stars on one half of the sky, does it mean that a thrown object will travel faster in that direction? If the object is at the edge of the universe, does it mean it will not be...

Ok so I was answering a question about a spacecraft accelerating through deep space with observer A in the front and observer B in the rear. I was asked to explain why a phton sent from A to B would appear blue shifted to B - simple enough I just plugged in a positive velocity to the...

Hello, I can answer some basic questions about GR, but I don't teach it, and I am not proficient in it. Yesterday a student asked a question that I felt I couldn't do justice to. Can you help? I had the student email the question so I could ask around. So far I haven't gotten any useful...

I can't figure out how to show that the Clausius entropy \Delta S= \int_{T_1}^{T_2} \frac{dQ}{T} is equivalent to the Boltzmann entropy S=kln \Omega at thermal equilibrium. Does anyone know how to do this?

One ton of Uranium used at a typical 1 GW reactor is equal to how many tons of coal being burned in a 1 GW coal fired plant?

Hi all. First post so please forgive the noobness. I have been listing to an audio book of "A briefer history of time" by that well known genital man and I have hit upon something I don't understand. My understanding of the Principle of Equivalence is that you will not be able to tell if you...

Hi all, There are a few doubts I hope I can get enlightened. If we were in a lab in outer space, free of gravitational influence and the lab accelerates downwards, what will we experience? do we 'move' to the top of the lab? What I'm confused about also is, why is that when you have an...

Hi, When I was just walking through the hallway of my department I found an exercise sheet asking the student to examine the following proof. Assume R\subset M\times M is a binary, symmetric, transitive relation. Then for any a,b \in M with a\sim _R b it follows by symmetry that b\sim _R a...

Homework Statement On set PxP, define (m,n)\approx(p,q) if m*q=p*n Show that \approx is an equivalence relation on PxP and list three elements in equivalence class for (1,2) Homework Equations The Attempt at a Solution I will appreciate any help how to start this problem...

Homework Statement Find relations that satisfying just Reflexive just Symmrtic just Transitive (R) & (S), but not (T) (R) & (T), but not (S) (S) & (T), but not (R) Homework Equations S=Z (a,b) \inR if <=> a>b (T) but, not (S) & (R). the ex is given in the class, but...

I'm curious about the famous formula, e=mc^2 . Since mass can be measured in any of several unit systems, shouldn't the formula read e\propto mc^2 ? Thanks!

Homework Statement question 1: Define ~ on Z by a ~ b if and only if 3a + b is multiple of 4. question 2: Let A = {1,2,3,4,5,6} and let S = P(A) (the power set of A). For a,b \in S define a ~ b if a and b have the same number of elements. Prove that ~ defines an equivalence...

Homework Statement Prove that when R and S are equivalence relations, then SR is equivalance relation when SR=RS. Homework Equations The Attempt at a Solution I want to know what sets should i take for the relations ? is it ok when i take R a a subset of X x Y and S as a...

This thread discusses an interesting point Jonathan Scott brought up at https://www.physicsforums.com/showthread.php?t=264782&page=2. A naive application of the equivalence principle (EP) would suggest that a free falling charge does not radiate, and that a charge on the surface of the Earth...

Please be nice to me, I'm new here. Anyway, help to solve this maths problem would be much appreciated: Homework Statement Work out a detailed proof (below) that the relation on the integers defined by p~q if and only if 7|p-q is an equivalence relation: a) the relation is reflexive b) the...

From equivalence principle, being on Earth is equivalent to being on lift accelerating at 9.8 m/s2...so if we are in a lift and that accelerates at 9.8 m/s2 then at some time we will cross the velocity of light.. my questions are 1) is Earth accelerating ? 2) what happens if the velocity...

Homework Statement Use the equivalence principle to explain the observation that a helium balloon leans forward in a forward-accelerating car. Homework Equations I don't think equations are needed. The Attempt at a Solution I'm completely confused on this question

Homework Statement There's this one exam problem that I cannot solve... Here it goes: Consider the set Z x Z+. Let R be the relation defined by the following: for (a,b) and (c,d) in ZxZ+, (a,b) R (c,d) if and only if ad = bc, where ab is the product of the two numbers a and b. a) Prove that...

Homework Statement A 20.00 ml stock sample of HBr is diluted to 50 mL. If 18.76 ml of 0.1345 M KOH was required to reach the equivalence point, what is the concentration of the stock HBr solution? Homework Equations KOH + HBr --> KBr + H20 M = conc. = mol./L n = V * conc. The...

Is the time dilation between two frames at different distances from a mass equivalent by some constant of proportionality to the time dilation between two frames in relative motion? For example: where r0 is the Schwarzschild radius and r is a radial coordinate greater than r0, this would be...

I was revising linear algebra and came across the topic of 'constructing complementary subspace given a subspace' - and since the proof (that used Zorn's lemma) of its (complementary subspace's) existence was not constructive, the author defined an equivalence relation in constructing a...

some of you will probly find this easy but this my first assignment after learning what a logical equivalence is so I can't warp my head around it. there are 2 problems I need help with. Homework Statement Find a compound proposition logically equivalent to p V q using only the logical...

My understanding is that, according to the Equivalence Principle, accelerated motion is indistinguishable from that of being in a gravitational field. So then, at the Event Horizon of a black hole, where gravity is too great for light to escape, wouldn't that be equivalent to accelerated motion...

For an weak acid/strong base titration, the pH at equivalence can be estimated by pH = pKa + pHtitrant/2. Can someone explain where that equation comes from? Thanks.

This is a really basic question that I don't know why I'm not getting. So from my understanding, a=b is defined as a=>b and b=>a. So say a^2=c and a=c^(1/2) so which implies which? Say I start from a=c^(1/2) I square both sides and I get a^2=c. So a=c^(1/2) => a^2=c. But if I...

Hi fellas I have been reading Road to Reality by Roger Penrose, but can't go beyond chapter 8. I do not understand why topological equivalence does not imply conformal equivalence. In particular I cannot really make sense of his argument as to why a thin torus is not conformally the same as a...

Use the standard logical equivalences to simlify the expression (-p^q)v-(pvq)... fanx folks!

I've been stuck with this problem: An ellipse can be defined as 1) locus of points for which is constant the sum of the distances from two fixed points (foci) 2) locus of points for which is constant the ratio between the distances from a fixed point (focus) and a fixed line (directrix)...

Homework Statement if w is the nth root of unity, i.e. w= exp(2pi/n i) show: (1-w)(1-w^2)...(1-w^{n-1})=n Homework Equations The Attempt at a Solution since w^(n-a)= complex congugate of w^a terms on the left hand side are going to pair up to give |1-w|^2 |1-w^2|^2... but I'm...

For the set A = {1,2,3,4,5,6,7}, determine whether script A is a partition of A. script A = {{1,3,},{5,6}, {2,4},{7}} Describe the partition for the equivalence relation T defined for x,y \in \mathbbc{R} by X T y iff \left[ \left[x \right] \right] = \left[ \left[y \right] \right] where...

Lagarias’ equivalence to the Riemann hypothesis should be discussed, i.e., if hn := n-th harmonic number := 1/1 + 1/2 + · · · + 1/n, and σn := divisor function of n := sum of positive divisors of n, then if n > 1, hn + ehn ln hn > σn. There is a $1,000,000 prize for the proof of this at...

I have a question... "Is the quotient set of a set S relative to a equivalence relation on S a subset of S?" I suppose "no",since the each member of the quotient set is a subset of S and consequently it is a subset of the power set of S,but I have e book saying that "yes",I am a bit...

Two clocks in a gravitational field separated by an altitude of x, exhibit constant time dilation. Two clocks accelerating in free space separated by the length x of their ship, exhibit non-constant time dilation. How is the principle of equivalence sustained when this fundamental measure of...

Homework Statement We define a relation ~ for N^2 by: (n, m) ~(k, l) <=> n + l = m + k Show that ~ is a equivalence relation Homework Equations A relation R on a set A is equivalent if R is: reflexive if x R x for all x that is an element of A symmetric if x R y implies y R x...

Q1. f is a continuous real valued function on [o,oo) and a is a real number Prove that the following statement are equivalent; (i) f(x)--->a, as x--->oo (ii) for every sequence {x_n} of positive numbers such that x_n --->oo one has that (1/n)\sum f(x_k)--->a, as n--->oo (the sum is taken...

Prove that if [i] and [j] are equivalence classes modulo 1. Prove that if [i] and [j] are equivalence classes modulo n such that [i]=[j], then gcd(i,n)=gcd(j,n) 2. Prove that if gcd(a,b)=1 and if c divides b, then gcd(a,c)=1. please help

Homework Statement Let G be a finitely generated abelian group and let T_p be the subgroup of all elements having order some power of a prime p. Suppose T_p \simeq \mathbb{Z}_{p^{r_1}} \times \mathbb{Z}_{p^{r_2}} \times \cdots \times \mathbb{Z}_{p^{r_m }} \simeq \mathbb{Z}_{p^{s_1}}...

Homework Statement if i have a polar equation: r=arctan(tan(x)) is that the same as: r=x ?? Homework Equations The Attempt at a Solution

One of my books defines a relation which is "evidently" an equivalence relation. It says that two idempotents in a ring P and Q are said to be equivalent if there exist elements X and Y such that P = XY and Q = YX. The proof that this relation is transitive eludes me. There is so little...

Homework Statement Let S be the set of integers. If a,b\in S, define aRb if ab\geq0. Is R an equivalence relation on S? Homework Equations The Attempt at a Solution Def: aRb=bRa \rightarrow ab=ba assume that aRb and bRc \Rightarrow aRc a=b and b=c since a=b, the substitute a...

x is a unit vector \in \Re^{2}. My textbook states that \frac{x}{||x||}=\frac{1}{||x||}x. What is the point of including \frac{1}{||x||}; why do they divide the vector by its length? Edit: I just looked at a book in Google's database, and from what I understand: e.g. \sqrt{{2^2+2^2+1^2}}=3...

So, here is the problem: Let x,y\in\mathbb{R}, R=\{{(x,y)\in\mathbb{R}^{2}|x= y r^{2}, for some r \in\mathbb{R}\}. Prove that R is an equivalence relation on \mathbb{R}. Relevant equations: R is an equivalence relation on \mathbb{R} if 1. (x,x)\in R for all x\in\mathbb{R} 2...

I've been using a vacuum system to filter solutions. I would like to move it over to a centrifuge. I have been having trouble equating the pressure in the vacuum to the required amount of force required by a centrifuge. Can someone help me out with the math? I'm looking a the vacuum pressure in...

Show equivalence of family of metrics on \Re^n: \rho^{(p)}:(x,y)\rightleftharpoons(\sum_{i=1}^{n}|x_i-y_i|^p)^{\frac{1}{p}} for p\geq1 The attempt at a solution Want to show for p,q\geq1, p\neq q, that \rho^{(p)} and \rho^{(q)} generate the same topology. I tried two methods: 1. Show that a...

So I'm kind of confused about the definition: a-b\in I Why a - b instead of a + b?

Einstien gives the example of an observer riding on the rim of a spinning disc. Using his length contracted rulers he measures the perimeter to be greater by a factor of gamma (y), than the 2\pi R he would have measured when the disc was stationary. When he measures the radius with the same...

In free space far removed from significant particulate matter, inertial reaction of a test mass will be isotropic. But if Mach's Principle is the root cause of inertia, then a nearby massive object should modify the inertia of such a test mass so that its reactance to acceleration will be...