Function Definition and 1000 Threads

Honestly not sure how to go about this. Again this is one equation of 4 that I have. I considered using Laplace transforms but taking the Laplace transform of a step function whose argument is one of the variables being solved for doesn't seem possible. Also, if there is an alternative way to...

Homework Statement:: Why is the heaviside function in the inverse laplace transform of 1? Relevant Equations:: N/A This is a small segment of a larger problem I've been working on, and in my book it gives the transform of 1 as 1/s and vice versa. But as I've looked online for help in figuring...

For a single particle, $$Z=\frac{1}{h^2}\int_{-\infty}^{\infty} e^{-\beta \frac{P^2}{2m}}d^2p \int e^{-U(r)}drd\theta= \frac{1}{h^2}(\frac{2\pi m}{\beta}) 2\pi [\int_{0}^{r_0}e^{U_0}dr+\int_{r_0}^{R}dr]$$ $$ =\frac{1}{h^2}(\frac{2\pi m}{\beta}) 2\pi [e^{U_0}(r_0)+(R-r_0)]=\frac{\pi...

The beginning is straight forward and I found f=x^2-2yz, which satisfies grad(f)=F. Then I calculated W= f(x,y,z)-f(0,1,1) since it's conservative. I get stuck when trying to find the max and mins. Given grad(f)=0 at extrema, we can see (0,0,0) is a point. On the boundary, I have to...

I am not sure, but since the partition function Z is just the sum of all Boltzmann Factor We can just add: (some terms don't appear in the image, by the way, the estimative is nice, the result is above ANS) But i didn't understand what the author did: While i didn't even care about the...

I would express the pendulum motion in form of polar coordinates with corresponding unit vectors in the x-z-plane (view images, ignore the german, sorry). How would you now bring in the circular movement, which is constantly changing its plane? Is it enough to simply add the representation of a...

Hey, please tell me if the following is correct. We have a continuous, increasing and strictly monotonic function on ##[a, b]##, and ##x_0\in[a,b]##. Let ##g(y)## be its inverse, and ##f(x_0)=y_0##. I want to show that ##|y-y_0|<\delta\implies|g(y)-g(y_0)|<\epsilon##. \begin{align*}...

In this paper, on quantum Ising model dynamics, they consider the Hamiltonian $$\mathcal{H} = \sum_{j < k} J_{jk} \hat{\sigma}_{j}^{z}\hat{\sigma}_{k}^{z}$$ and the correlation function $$\mathcal{G} = \langle \mathcal{T}_C(\hat{\sigma}^{a_n}_{j_n}(t_n^*)\cdot\cdot\cdot...

Obviously, a priori it is not possible tu use the Taylor series because the derivative ##\sim (x-1)^{1/n-1}## is not well defined in x=1. Is there any mathematical trick? or, other approximation?

could anyone explain why in the page of book this figure is related to hartree-fock? I mean why if t1>t2 we have these possibilities? and why not particle propagate from x2t2 to x3t3 instead x3t3+?

h(x) = 0 for x ≤ 0 h(x) = x^2 for x>0 But my book says h(x) = 0 for x<0 h(x) = x^2 for x≥0 Can my solution (the first one) work as well? Because the actual function value at x = 0 is zero. I feel like my solution is more elegant.

I have to show that $\forall z\in B(0,0.4)$, there exists an $x\in B(0,1)$ such that $f(x)=z$ but I am not sure how to show this. From the reverse triangle inequality $$-|f(x)-f(y)|+|x-y|\leq 0.1|x-y|\implies |f(x)-f(y)|\geq 0.9|x-y|$$ im not sure if this helps.

Is it possible to find three quadratic polynomials $f(x),\,g(x)$ and $h(x)$ such that the equation $f(g(h(x)))=0$ has the eight roots 1, 2, 3, 4, 5, 6, 7 and 8?

$$Q_{(\alpha, \beta)} = \sum_{N=0}^{\infty} e^{\alpha N} Z_{N}(\alpha, \beta) \hspace{1cm} (3.127)$$ Where ##Q## is the grand partition function, ##Z_N## is the canonical partition function and: $$\beta = \frac{1}{kT} \hspace{1cm} \alpha = \frac{\mu}{kT} \hspace{1cm} (3.128)$$ In the case of an...

I have a basic question in elementary quantum mechanics: Consider the Hamiltonian $$H = -\frac{\hbar^2}{2m}\partial^2_x - V_0 \delta(x),$$ where ##\delta(x)## is the Dirac function. The eigen wave functions can have an odd or even parity under inversion. Amongst the even-parity wave functions...

Hello all, This is a problem of a different flavour from my usual shenanigans. I'm looking at a function $$f(m,n)=\frac{m^2n^2}{(m+n)(m-n)}$$ and am trying to determine if there are any two pairs of values ##(m_1,n_1)## and ##(m_2,n_2)## which evaluate to the same result. Assume that...

Practically it is said that, given two spin states |u⟩ (up) and |d⟩ (down) - which are the spin measured along the +z and -z semiaxes - such that they are orthogonal ( ⟨u|d⟩ = ⟨d|u⟩ = 0), it is possible to write any other spin states using a linear combination of these two (because they are a...

I have a formula for cost calculation that contain x and y two variable. I have to find the value of (x,y) where that formula will gives minimum value as cost should not be equal to zero, it has some minimum value. I took 1st partial derivative with respect to x and then with y and found the...

This seems like it should be an easy and obvious thing to look up, but I had the hardest time finding it. Is there any graph which shows, as I increase the beam energy of a particle accelerator, what particles can be produced at each energy? Just looking for something ballpark here. Obviously...

I don't see why it is not ##P(\omega)\propto |\langle \psi | \mathbb{P}_{\omega}|\psi\rangle |^2.## After all, the wavefunction ends up collapsing from ##|\psi\rangle## to ##\mathbb{P}_{\omega}|\psi\rangle.##

Greetings, similar to my previous thread (https://www.physicsforums.com/threads/lennard-jones-potential-and-the-average-distance-between-two-particles.990055/#post-6355442), I am trying to calculate the average inter-particle distance of particles that interact via Lennard Jones potentials...

Prove that if $[a,\,b]\subset \left(0,\,\dfrac{\pi}{2}\right)$, $\displaystyle \int_a^b \sin x\,dx>\sqrt{b^2+1}-\sqrt{1^2+1}$.

My son asked me to look at his Logitech Z-2300 system (Amp, sub-woofer and 2 satellite speakers) that was acting up. He had already taken a lot of it apart to see if he could find a loose connection. In checking things out, I found this part in with all the screws and loose hardware. It's the...

This is really a simple question, but I'm stuck. Suppose we have a function ##\vartheta'(\vartheta) = \vartheta## and that ##\vartheta = \vartheta(\varphi)## and we know what ##\vartheta(\varphi)## is. How should I view ##\frac{\partial \vartheta'}{\partial \varphi}##? Should I set it equal to...

My textbook says ##\vec r (\theta) = r \hat r (\theta)##, where ##\hat r (\theta)## is the terminal arm (a position vector in some sense). It can be seen that both ##\vec r (\theta)## and ##\hat r (\theta) ## are function of ##\theta##; whereas, the length of the vector ##r## is not. I...

First, I try to define the function in the figure above: ##V(t)=100\left[sin(120\{pi}\right]##. Then, I use the fact that absolute value function is an even function, so only Fourier series only contain cosine terms. In other words, ##b_n = 0## Next, I want to determine Fourier coefficient...

Should I just follow the original question? If given as ##f(x)=\ln x^4## then the domain is x ∈ ℝ , x ≠ 0 and if given as ##f(x) = 4 \ln x## the domain is x > 0? So for the determination of domain I can not change the original question from ##\ln x^4## to ##4 \ln x## or vice versa? Thanks

$f: \mathbb{R^2} \rightarrow \mathbb{R}$, $f(x,y) = x^2+y^2-1$ $X:= f^{-1} (\{0\})=\{(x,y) \in \mathbb{R^2} | f(x,y)=0\}$ 1. Show that $f$ is continuous differentiable. 2. For which $(x,y) \in \mathbb{R^2}$ is the implicit function theorem usable to express $y$ under the condition $f(x,y)=0$...

Let $$Y(t)=tanh(ln(1+Z(t)^2))$$ where Z is the Hardy Z function; I'm trying to calculate the pedal coordinates of the curve defined by $$L = \{ (t (u), s (u)) : {Re} (Y (t (u) + i s (u)))_{} = 0 \}$$ and $$H = \{ (t (u), s (u)) : {Im} (Y (t (u) + i s (u)))_{} = 0 \}$$ , and for that I need to...

When wave function collapses how long is it collasped... Shooting electrons at a double slit and observing the electrons before they reach the 2 slits collasped the wave function...so is its behavior particle like forever? Quantum mechanics is simple however wrapping ones head around it is...

I have the following probability density function: $$f(x) =...

The function $f$ is defined on the set of integers and satisfies \[ f(n)=\begin{cases} n-3, & \text{if} \,\,n\geq 1000 \\ f(f(n+5)), & \text{if}\,\, n< 1000 \end{cases} \] Find $f(84)$.

I would differentiate this twice and plug it into the S.E, but for that I'll need E. Which I don't have. Please provide me some direction.

If the function is not differentiable at point. Can we consider this point is critical point to the function? f(x) = (x-3)^2 when x>0 = (x+3)^2 when x<0 he asked for critical points in the closed interval -2, 2

I attached a file with some explanations of the variables in the code and the plot that I should get. I don't know what is wrong. Any help will appreciated. from scipy.integrate import quad import numpy as np from scipy.special import gamma as gamma_function from scipy.constants import e...

My question and solution that I've tried out are in attachment. Is it true my steps?

Suppose ##\nu## is a measure on some ##\sigma##-algebra ##\mathcal{A}##. Then we must have for all ##A \in \mathcal{A}## either ##A## or ##A^c## is finite, but not both. Because otherwise ##\nu(A)## is undefined or not well defined. I've verified that ##\lbrace \emptyset, X \rbrace## and...

Suppose I'm solving $$y''(t) = x''(t)$$ where $$x(t)$$ is the ramp function. Then, by taking the Laplace transform of both sides, I need to know $x'(0)$ which is discontinuous. What is the appropriate technique to use here?

I've tried with this work in attachment. i&m not sure of my answer is correct.

Hello! I have been recently studying Quantum mechanics alone and I've just got this question. If the potential function V(x) is an even function, then the time-independent wave function can always be taken to be either even or odd. However, I found one case that this theorem is not applied...

Summary:: Abstract algebra I have a problem with this task. Please help. [Moderator's note: Moved from a technical forum and thus no template.]

From my understanding of diffraction pattern is supposed to result in something like this However when I plot it I get the central peak without the ripples (even when broadening the view). My result My code is as follows %1) Define the grid. Define vectors so that they include 0...

Hello, Please I need help to find the objective function of a linear program (attachement : example). I tried to figure it out from the formula provided in (attachement : formula) but I couldn't understand it, it's written (MIN(lambda)wj) I think it's the key to resolve my question ! ( Full file...

Hello, Please I need help to find the objective function of a linear program (attachement : example). I tried to figure it out from the formula provided in (attachement : formula) but I couldn't understand it, it's written (MIN(lambda)wj) I think it's the key to my question ! ( Full file is...

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 ...

Hello everyone, can anybody help me with this problem? The solution is for all odd prime numbers, but I have no idea how to solve it. Any help will be greatly appreciated.

I have the following probability density function (in Maple notation): f (x) = (1 / ((3/2) * Pi)) * (sin (x)) ** 2 with support [0; 3 * Pi] Now I want to transform x so that 0 -> (3/2) * Pi and 3 * Pi -> (15/2) * Pi and the new function is still a probability density function. How should I...

shown in attachment

Para f (θ) = √3.cos² (θ) + sen (2θ), uma inclinação da reta tangente, uma função em θ = π / 6, é?

Transformed circuit: Using KVL, Now, I am unsure about the current to use KVL in this case. As far as equation goes: Vi(s) =(I1*R)+(I3*R)+Vc(s), where Vc(s) = V0(s)/u as shown in the circuit. How am I supposed to find the current I1 and I3 for the two resistors in this case? Thanks