Theorem Definition and 1000 Threads

In this exercise, we consider simple, nonsymmetric random walk. Suppose 1/2 < q < 1 and ##X_1, X_2, \dots## are independent random variables with ##\mathbb{P}\{X_j = 1\} = 1 − \mathbb{P}\{X_j = −1\} = q.## Let ##S_0 = 0## and ##S_n = X_1 +\dots +X_n.## Let ##F_n## denote the information...

I'm really struggling with this one. A newbie to using the residue theorem. I'm trying to solve this by factorising the denominator to find values for z0 and I have: ##z=\frac{-\sqrt{2}+i\sqrt{2}}{2}## and ##z=\frac{-\sqrt{2}-i\sqrt{2}}{2}## I also know that sin(3π/8)=...

My thought was to break up the sentence into its equivalent form: (A ->~~A) & (~~A -> A) From there I assumed the premise of both sides to use indirect proofs, so: 1. ~(A -> ~~A) AP 2. ~(~A or ~~A) 1 Implication 3. ~~A & ~~~A 2...

Alfvén’s theorem is very famous in plasma physics. It is also often used in astrophysics. The link in Wiki: https://en.wikipedia.org/wiki/Alfv%C3%A9n%27s_theorem However, after a series of continuous reasoning, it seems that this theorem has problem. What errors can be hidden in the reasoning...

But if I would assume that the efficiency of the carnot's engine is greater than the other engine and the carnot engine is driving the other engine backward as a refrigerator ,that would lead to the same contradiction hence disproving carbot's theorem! Is there something wrong I have done...

I have been trying to solve the following problem: Point-like object at (0,0) starts moving from rest along the path y = 2x2-4x until point A(3,6). This formula gives the total force applied on the object: F = 10xy i + 15 j. a) Find the work done by F along the path, b) Find the speed of the...

I'm referring to this result: But I'm not sure what happens if I apply a linear differential operator to both sides (like a derivation ##D##) - more specifically I'm not sure at what point should each term be evaluated. Acting ##D## on both sides I'll get...

Hi, I am unfortunately stuck with the following task I started once with the hint that at very low temperatures the diatomic ideal gas behaves like monatomic gas and has only three degrees of freedom of translation ##f=3##. If you then excite the gas by increasing the temperature, you add two...

There has been a lot of discussion on Bell's theorem here lately. Superdeterminism as a Bell's theorem loophole has been discussed extensively. But I have not seen discussion about Karl Hess, Hans De Raedt, and Kristel Michielsen's ideas, which essentially suggest that there are several hidden...

The work-energy theorem is the connection between expressing mechanics taking place in terms of force-and-acceleration, ##F=ma## and representing mechanics taking place in terms of interconversion of kinetic energy and potential energy. The following statements are for the case that there is a...

Here is Lars Olsen's proof. I'm having difficulty in understanding why ##y## will lie between ##f_a (a)## and ##f_a(b)##. Initially, we assumed that ##f'(a) \lt y \lt f'(b)##, but ##f_a(b)## doesn't equal to ##f'(b)##.

The expression ##\langle \cal H \rangle_k## is the expected value of the canonical ensemble. The Hamiltonian is defined as follows, with the scaling ##\lambda## ##\lambda \cal H ## : ##\lambda H(x_1, ...,x_N)=H(\lambda^{a_1}x_1,....,\lambda^{a_N}x_N)## As a hint, I should differentiate the...

I considered the vector field ##a \times F##, and applied Stoke's theorem. I obtained that $$\int_C (a \times F) \cdot dr = \int_C (F \times dr ) \cdot a.$$ Now, $$\nabla \times (a \times F) = a (\nabla \cdot F) - (a \cdot \nabla) F.$$ Using Stoke's theorem for the vector field ##a \times F##...

Consider a convex shape ##S## of positive area ##A## inside the unit square. Let ##a≤1## be the supremum of all subsets of the unit square that can be obtained as disjoint union of finitely many scaled and translated copies of ##S##. Partition the square into ##n×n## smaller squares (see...

Hi, unfortunately, I am not getting anywhere with task b In the lecture we had the special case that ##\vec{M}=0## , ##I_x=I_y=I , I \neq I_z## and ##\omega_z=const.## Then the Euler equation looks like this. $$I_x\dot{\omega_x}+\omega_y \omega_z(I_z-i_y)=0$$ $$I_y\dot{\omega_y}+\omega_z...

Hello! If I have a 2 level system, with the energy splitting between the 2 levels ##\omega_{12}## and an external perturbation characterized by a frequency ##\omega_P##, if ##\omega_{12}>>\omega_P## I can use the adiabatic approximation, and assume that the initial state of the system changes...

Calculate surface integral ## \displaystyle\iint\limits_S curl F \cdot dS ## where S is the surface, oriented outward in below given figure and F = [ z,2xy,x+y]. How can we answer this question?

I know of Bell's theorem. Kochen-Specker theorem is supposed to be a complement to Bell's theorem. I tried to understand it by reading the Wikipedia article. But I couldn't fully grasp the essential feature of this theorem, in what way it complements Bell's theorem. What are the main...

##curl([x^2z, 3x , -y^3],[x,y,z]) =[-3y^2 ,x^2,3]## The unit normal vector to the surface ##z(x,y)=x^2+y^2## is ##n= \frac{-2xi -2yj +k}{\sqrt{1+4x^2 +4y^2}}## ##[-3y^2,x^2,3]\cdot n= \frac{-6x^2y +6xy^2}{\sqrt{1+4x^2 + 4y^2}}## Since ##\Sigma## can be parametrized as ##r(x,y) = xi + yj +(x^2...

Are my answers to a and b correct? a) In a three-dimensional situation, the spatial variation of a scalar field is given by the gradient. What is the one-dimensional counterpart? Answer:The derivative b) In a three-dimensional situation, a volume integral of a divergence of a vector field can...

What geometry theorem is used to be able to state that 8/4 = x/6 ??

The following properties of big-O notation follow from the definition: (i) if ##f(x)=O(u(x))## as ##x\rightarrow{a}##, then ##Cf(x)=O(u(x))## as ##x\rightarrow{a}## for any value of the constant ##C##. (ii) If ##f(x)=O(u(x))## as ##x\rightarrow{a}## and ##g(x)=O(u(x))## as ##x\rightarrow{a}##...

My answer:

We have to prove: If ##f: [a,b] \to \mathcal{R}## is continuous, and there is a ##L## such that ##f(a) \lt L \lt f(b)## (or the other way round), then there exists some ##c \in [a,b]## such that ##f(c) = L##. Proof: Let ##S = \{ x: f(x) \lt L\}##. As ##S## is a set of real numbers and...

The theorem is pretty clear...out of curiosity i would like to ask...what if we took ##n-3## factors...then the theorem would not be true because we shall have; ##[n!=6⋅4⋅5⋅6 ... n]## and ##[2^n = 8⋅2⋅2⋅2 ...2]## What i am trying to ask is at what point do we determine the number of terms...

The last formula is what I was going for, since it arises as the momentum flux in fluid dynamics, but in the process I came across the rest of these formulas which I’m not sure about. The second equation is missing a minus sign (I meant to put [dA X grad(f)]). Are they correct? Do they have...

If correct, as non-physicist, I wonder why the vast jump to "spooky action" is seen as more plausible as some new type of particle faster than the speed of light. Consider the time long before the discovery of radio communication, how weird it must have been to theorize about that. The speed of...

Basically surface B is a cylinder, stretching in the y direction. Surface C is a plane, going 45 degrees across the x-y plane. Drawing this visually it's self evident that the normal vector is $$(1, 1, 0)/\sqrt 2$$ Using stokes we can integrate over the surface instead of the line. $$\int A(r)...

Solution Write the Taylor formula for ##e^x## at ##x=0##, with ##n## replaced by ##2n+1##, and then rewrite that with ##x## replaced with ##-x##. We get: $$e^x=1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\cdots+\dfrac{x^{2n}}{(2n)!}+\dfrac{x^{2n+1}}{(2n+1)!}+O(x^{2n+2})$$...

I would like to check my understanding here to see if it is correct as I am currently stuck at the moment. From the question, I can gather that: P(Rain | Dec) = 9/30 P(Cloudy | Rain) = 0.6? P(Cloudy | Rain) = 0.4 To answer the question: P(Rain | <Cloudy, Morning, December> ) = P(Rain) *...

I parameterize surface A as: $$A = (2cos t, 0, 2sin t), t: 0 \rightarrow 2pi$$ Then I get y from surface B: $$y = 2 - x = 2 - 2cos t$$ $$r(t) = (2cost t, 2 - 2cos t, 2sin t)$$ Now I'm asked to integral over the surface, not solve the line integral. So I create a new function to cover the...

I read about Noether's theorem that says how for every symmetry there is a conserved quantity. Seems kind of obvious. Does anyone understand it well enough that they can explain precisely why that notion is profound?

What is the main reason for a free field staying free according to Haag's theorem?

Hello, I am currently working on the proof of Minkowski's convex body theorem. The statement of the corollary here is the following: Now in the proof the following is done: My questions are as follows: First, why does the equality ##vol(S/2) = 2^{-m} vol(S)## hold here and second what...

I am trying to solve the given question based on energy conservation,but am stuck with the analysis of the equations. The question says find the velocity of the bigger block when the smaller block initially given a velocity v and sliding on the horizontal part of the bigger block reaches the...

Consider modulo ## 10 ##. Then ## 10=5\cdot 2 ##. Applying the Fermat's theorem produces: ## 3^{4}\equiv 1\pmod {5} ##. This means ## (3^{4})^{25}=3^{100}\equiv 1\pmod {5} ##. Observe that ## 3\equiv 1\pmod {2}\implies 3^{100}\equiv 1\pmod {2} ##. Now we have ## 5\mid (3^{100}-1) ## and ##...

Consider a certain integer between ## 1 ## and ## 1200 ##. Then ## x\equiv 1\pmod {9}, x\equiv 10\pmod {11} ## and ## x\equiv 0\pmod {13} ##. Applying the Chinese Remainder Theorem produces: ## n=9\cdot 11\cdot 13=1287 ##. This means ## N_{1}=\frac{1287}{9}=143, N_{2}=\frac{1287}{11}=117 ## and...

prove: The 2nd axiom of mathematical logic 2) $((P\implies(Q\implies R))\implies((P\implies Q)\implies(P\implies R))$ By using only the deduction theorem

do these proposals violate the Coleman-Mandula theorem since they combine space-time and internal symmetries via SU(2)r and SU(2)l how plausible are these proposals as actual physical theories that unify gravity with SU(2) weak force? Graviweak Unification F. Nesti, R. Percacci...

My question is, if the determinism theorem is a good explanation, which covers all holes of the entanglement experiment. why are people still concluding its a 'spooky' superposition which is only determined by a measure and then somehow affects the other measurement. What am I missing? Why is...

Hello physics researchers, teachers and enthusiasts. I notice one little thing is confusing me in the derivation of Bernoulli's equation in the article, they write:$$dW = dK + dU$$where dW is the work done to the fluid, dK is the change in kinetic energy of the fluid, and dU is the change in...

According to Helmholtz’s theorem, if electric charge density goes to to zero as r goes to infinity faster than 1/r^2 I'm able to construct an electrostatic potential function using the usual integral over the source, yet I don't understand how this applies to a chunk of charge in some region of...

I am a little confused with the Poynting theorem https://en.wikipedia.org/wiki/Poynting%27s_theorem . When we use this equation, the energy density that enters in $$\partial u / \partial t$$ is the one due only to the fields generated by charges/source itself? That is, if we have a magnetic...

Hi, I am looking for a formal proof of Thevenin theorem. Actually the first point to clarify is why any linear network seen from a port is equivalent to a linear bipole. In other words look at the following picture: each of the two parts are networks of bipoles themselves. Why the part 1 -- as...

This video demonstrates the Dzhanibekov effect (instability when spinning arround the intermediate axis). In order to achieve the best results, is it better for the three MoI's to be close together, or for them to have widely differing values?

(OBS: Don't take the index positions too literal...) Generally it is easy to deal with these type of exercises for discrete system. But since we need to evaluate it for continuous, i am a little confused on how to do it. Goldstein/Nivaldo gives these formulas: I am trying to understand how...

Greetings the solution is the following which I understand I do understand why the current orientation of the Path is positive regarding to stocks (the surface should remain to the left) but I don´t understand why the current N vector of the surface is positive regarding stockes theorem...

For the second uniqueness theorem of electrostatics to apply, does the outer boundary enclosing all the conductors have to be at a constant potential?

My attempt; ##4x^3+kx^2+px+2=(x^2+λ^2)(4x+b)## ##4x^3+kx^2+px+2=4x^3+bx^2+4λ^2x+bλ^2## ##⇒k=b, p=4λ^2 , bλ^2=2## ##\dfrac{4λ^2}{bλ^2}=\dfrac{p}{2}## ##\dfrac{4}{b}=\dfrac{p}{2}## ##⇒8=pb## but ##b=k## ##⇒8=kp## Any other approach appreciated...