Theorem Definition and 1000 Threads

Goldstein 2nd ed. In its Appendix is given the derivation of Bertrands Theorem.Here ##x=u-u_0## is the deviation from circularity and ##J(u)=-\frac{m}{l^{2}} \frac{d}{d u} V\left(\frac{1}{u}\right)=-\frac{m}{l^{2} u^{2}} f\left(\frac{1}{u}\right)## If the R.H.S of A-10 was zero, the solution...

Wikipedia on Bertrands theorem, when discussing the deviations from a circular orbit says: >..."The next step is to consider the equation for ##u## under small perturbations ##{\displaystyle \eta \equiv u-u_{0}}## from perfectly circular orbits" (Here ##u## is related to the radial distance...

Using the transformation for ##t##, I obtained $$\mathrm{d}t'=\left(1+\frac{\partial f}{\partial t}\right)\mathrm{d}t+\frac{\partial f}{\partial r}\mathrm{d}r$$. Using this equation, I substituted it into the general line element to obtain \begin{align*}...

Preceding Problem. Let ##y=f(x)## be a continuous function defined on a closed interval ##[0, b]## with the property that ##0 < f(x) < b## for all ##x## in ##[0, b]##. Show that there exist a point ##c## in ##[0, b]## with the property that ##f(c) = c##. This problem can be solved by letting...

So I've seen other threads on here with the same problem from a few years ago, and I'm just not getting the same answers. However, I followed along with a similar problem in the textbook and used all the same methods, so can't understand where I've gone wrong, or if I even am wrong. Also not...

Mihăilescu's theorem proves that Catalan's conjecture is true. That is for x^a - y^b = 1, the only possible solution in naturual numbers for this equation is x=3, a=2, y=2, b=3. What is not clear to me is this. Does Mihăilescu's theorem prove that the difference between any other two...

Theorem: Let ## f(x), g(x) \in \mathbb{F}[ x] ## by polynomials, s.t. the degree of ## g(x) ## is at least ## 1 ##. Then: there are polynomials ## q(x), r(x) \in \mathbb{F}[ x] ## s.t. 1. ## f(x)=q(x) \cdot g(x)+r(x) ## or 2. the degree of ## r(x) ## is less than the degree of ## g(x) ## Proof...

Hi, I was attempting the following question and just wanted to check whether my working was correct: Question: A bag has three coins in it which are visually indistinguishable, but when flipped, one coin has a 10% chance of coming up heads, another as a 30% chance of coming up heads, and the...

If you go to "The Abel Prize Interview 2016 with Andrew Wiles" on YouTube, there is a statement made by Andrew Wiles beginning at about 4:10 and ending about 4:54 where he mentions there are some new number systems possible where the fundamental theorem of arithmetic does not hold. I don't...

A simulation/animation/explanation based on the inertial frame only: The previous videos referenced there are here: See also this post for context on the Veritasium video: https://mathoverflow.net/a/82020 Note to mods: The previous thread is not open anymore so I opened a new one. Feel free...

Given surface ##S## in ##\mathbb{R}^3##: $$ z = 5-x^2-y^2, 1<z<4 $$ For a vector field ##\mathbf{A} = (3y, -xz, yz^2)##. I'm trying to calculate the surface flux of the curl of the vector field ##\int \nabla \times \mathbf{A} \cdot d\mathbf{S}##. By Stokes's theorem, this should be equal the...

Here I'm asking solely about the circle pictograms. Please eschew referring to, or using, numbers as much as possible. Please explain using solely the circle pictograms. Undeniably, I'm NOT asking about how to divide numbers. I don't understand 1. How do I "visually" divide Circle 1...

I saw this question at https://math.codidact.com/posts/283253.

I recently learned how to calculate the centroid of a semi-circular ring of radius ##r## using Pappus's centroid theorem as ##\begin{align} &4 \pi r^2=(2 \pi d)(\pi r)\nonumber\\ &d=\frac {2r}{\pi}\nonumber \end{align}## Where ##d## is the distance of center of mass of the ring from its base...

1. For regions that contain charge density, does the 1st uniqueness theorem still apply? 2. For regions that contain charge density, does the 'no local extrema' implication of Laplace's equation still apply? I think not, since the relevant equation now is Poisson's equation. Furthermore...

The correct answer is ##\frac{\pi a^2 h} 2## by using the standard approach. However when I tried using the divergence theorem to solve this problem, I got a different answer. My work is as follows: $$\iint_S \vec F\cdot\hat n\, dS = \iiint_D \nabla\cdot\vec F\,dV$$ $$= \iiint_D \frac{\partial...

It is more or less a generic problem of stokes theorem: ##\int_{\gamma} F dr##, where ##F = (-y/(x²+y²) + z,x/(x²+y²),ln(2+z^10))## and gamma is the intersection of ##z=y^2, x^2 + y^2 = 9## oriented in such way that its projection in xy is traveled clockwise. So, i decided to apply stokes...

I screenshot https://math.stackexchange.com/q/4185842.

The Noether's theorem for finite Hamiltonian systems says that: My question is: If I know a symmetry how can I write the first integral?

Sorry if there's latex errors. My internet connection is so bad I can't preview. Here's the wikipedia proof I'm referring to. I'm fine with the steps up to $$W(x,0) = W_0 (x) [1 + \beta f_0 (x(0) - \langle x \rangle_0) ]$$ where ##W(x,t)## is the probability density of finding the system at...

Greetings! here is the following exercice I understand that when we follow the traditional approach, (prametrization of the surface) we got the answer which is 8/3 But why the divergence theorem can not be used in our case? (I know it's a trap here) thank you!

I just wanted to point out a resource useful for dealing with claims of violating Bell's theorems. You can point the claimant at https://algassert.com/quantum/2015/10/11/Bell-Tests-vs-No-Communication.html and say "I won't believe you unless you can make the 'Write Your Own Classical CHSH...

The articles: https://lmcs.episciences.org/5328/pdf http://argo.matf.bg.ac.rs/publications/2013/2013-icga-krk-sat.pdf http://archive.ceciis.foi.hr/app/public/conferences/1/papers2012/dkb3.pdf KRK endgame is a win for white regardless of starting position, with the trivial drawing exception in...

Hi, In Chapter 5 Munkres proves the Tychonoff Theorem and after proving the theorem the first exercise is: Let ##X## be a space. Let ##\mathcal{D}## be a collection of subsets of ##X## that is maximal with respect to finite intersection property (a) Show that ##x\in\overline{D}## for every...

I don't know how to solve this proof Prove that a set $M \subset C([a, b])$ for which there exist $m. L> 0$ and $x_0 \in [a; b]$ such that $|f(x_0)| \leq{} m$ for all $f \in M$ and $|f(x)-f(y)| \leq{} L |x-y|$ for all $f\in M$ and for all $x,y \in [a,b]$ is relatively compact in $C([a, b])$ My...

I’m looking over a recent paper mentioned in another thread. It claims to refute Bell’s theorem. At first glance, the model presented in the paper doesn’t appear consistent with QM. Here’s a simple example. Suppose we set both polarizers to the same angle ##\alpha = \pi /4##. In the model...

I hate to create a thread for a step in a calculation, by I don't know what else to do. I'm having a lot of trouble reproducing E. Weinberg's index calculation (found here https://inspirehep.net/literature/7539) that gives the dimension of the moduli space generated by BPS solutions in the...

The Paper “On a contextual model refuting Bell’s theorem” has now been published by the journal EPL (Europhysics Letters) and is available under https://iopscience.iop.org/article/10.1209/0295-5075/134/10004 In this paper a contextual realistic model is presented which correctly predicts...

My research leads to 2 slightly different equations. See equations 1 and 2 attached. Also, for equation 1 should Wext be Wnet ext ?

Hi, I have a question, or am looking for clarification, about the no-cloning theorem and state tomography. My understanding is that the theorem states one cannot make an exact copy of a quantum state. I was also reading about state state tomography where it was said* 'On the other hand, the...

Guys, I have a problem that really needs you guys to help, I know it is a stupid question but please bear with me: Context: You have a block on a slope(has friction) you use a string to pull the block up with constant speed. Problem: So according to the network theorem, the work net is equal...

If ##f## is a constant function, then choose any point ##x_0##. For any ##x\in K##, ##f(x_0)\geq f(x)## and there is a point ##x_0\in K## s.t. ##f(x_0)=\sup f(K)=\sup\{f(x_0)\}=f(x_0)##. Now assume that ##f## is not a constant function. Construct a sequence of points ##x_n\in K## as follows...

I'm thinking I should apply Thevenin's Theorem to find the voltage, but I need to find I_D in order to be able to calculate U_S.

The Earnshaw’s theorem comes directly from Maxwell equation so it should be unavoidable in any classical situation. The theorem usually disallows magnetic levitation. However, there are loopholes. Quoting wikipedia "Earnshaw's theorem has no exceptions for non-moving permanent ferromagnets...

I'm trying to prove Plancherel's theorem for functions $$f\in L^1\cap L^2(\mathbb{R})$$. I've included below my attempt and I would really appreciate it if someone could check this for me please, and give me any feedback they might have. **Note:** I am working with a slightly different...

Loosely speaking, the Bell theorem says that any theory making the same measurable predictions as QM must necessarily be "nonlocal" in the Bell sense. (Here Bell locality is different from other notions of locality such as signal locality or locality of the Lagrangian. By the Bell theorem, I...

(1) We want to find the voltage across ##R_L## (2) We remove the load and label the terminals ##V_T## (3) The equivalent network of (2) So basically the voltage across the load is ##V_{th}## but when we find the equivalent network we put that same voltage behind a Thevenin...

I attached a screenshot of the book (sorry no pdf available for this book). Right above the somewhat central line they give the theorem that if there are m currents and n nodes, then there will be n - 1 independent equations from the current law and m - n - 1 from the voltage law. I count 4...

The Fundamental Theorem of Arithmetic essentially states that any positive whole number n can be written as: ##n = p_1^{a_1} \cdot p_2^{a_2} \cdot p_3^{a_3} \cdot \dots## where ##p_1##, ##p_2##, ##p_3##, etc. are all the primes, and ##a_1##, ##a_2##, ##a_3##, etc. are non-negative integers...

Use the Squeeze Theorem to find the limit. lim [x^2 • (1 - cos(1/x)] x--> 0 Let me see. -1 ≤ cos (1/x) ≤ 1 -x^2 ≤ x^2 • [1 - cos(1/x)] ≤ x^2 -|x^2| ≤ x^2 • [1 - cos(1/x)] ≤ |x^2| lim -|x^2| as x tends to 0 = 0. lim |x^2| as x tends to 0 = 0. . By the Squeeze Theorem, [x^2 • (1 -...

Use the Squeeze Theorem to find the limit. lim (x^2 • sin(1/x)) x--> 0 Let me see. -1 ≤ sin (1/x) ≤ 1 -x^2 ≤ x^2 • sin(1/x) ≤ x^2 -|x^2| ≤ x^2 • sin(1/x) ≤ |x^2| lim -|x^2| as x tends to 0 = 0. lim |x^2| as x tends to 0 = 0. . By the Squeeze Theorem, x^2 • sin(1/x) was squeezed between...

If 0 ≤ f(x) ≤ 1 for every x, show that lim [x^2 • f(x)] = 0. x--> 0 Let me see. 0 ≤ f(x) ≤ 1 Multiply all terms by x^2. 0 • x^2 ≤ x^2• f(x) ≤ 1 • x^2 0 ≤ x^2 • f(x) ≤ x^2 Is this right so far? If correct, what's next?

Proof goes like this: (1) Prove the existence of open intervals centered around the end-points of the domain such that the image of the points in these intervals through ##f## has the same sign as the image of the end-point through ##f##. In other words, prove that there is a ##\delta>0## such...

Explain why the Intermediate Value Theorem gives no information about the zeros of the function f(x) = ln(x^2 + 2) on the interval [−2, 2]. Let me see. Let x = -2. f(-2) = ln((-2)^2 + 2) f(-2) = ln(4 + 2) f(-2) = ln (6). This is a positive value. When I let x be 2, I get the same answer...

So I've just learned Norton's Theorem and I have this problem on my homework assignment that is wrong. I've checked the answer with a circuit simulator(PSPICE) and the simulation said that V0 should be a drop of 2V. However, my simplified circuit shows a voltage drop of 4V. I have been staring...

I wanted to ask about a step I couldn't understand in Tong's notes$$\int_M d^n x \partial_{\mu}(\sqrt{g} X^{\mu}) = \int_{\partial M} d^{n-1}x \sqrt{\gamma N^2} X^n = \int_{\partial M} d^{n-1}x \sqrt{\gamma} n_{\mu} X^{\mu}$$we're told that in these coordinates ##\partial M## is a surface of...

In Elementary Geometry we can use drawing figure to guess the geometry theorem.How can we guess a theorem in Math in general?

Hello at all! I have to solve this exercise: A tampon diagnostic test provides 1% positive results. The positive predictive values (probabilities of positive test disease) and negative (absence disease given negative test) are respectively 0.95 and 0.98. What is the prevalence of the disease...

I found three projection operators $$P_{1}= \begin{pmatrix} 1/2 & & \\ & -\sqrt{2}/2 & \\ & & 1/2 \end{pmatrix}$$ $$P_{2}= \begin{pmatrix} 1/2 & & \\ & \sqrt{2}/2 & \\ & & 1/2 \end{pmatrix}$$ $$P_{3}= \begin{pmatrix} -1/\sqrt{2} & & \\ & & \\ & & 1/\sqrt{2} \end{pmatrix}$$ From this five...

I have tried many times to solve this network, but can't understand how to get current in each resistors by superposition theorem. Please help me to solve and find currents in each 3 resistors with solution. Note:- The figure is attached below.