Function Definition and 1000 Threads

Let $X$ be a topological space and ##Y## a set. A function ##f: X \to Y## is said to be locally constant if, for every ##x \in X##, there is an open set ##U## containing ##x## so that the restriction ##f|_U: U \to Y## is constant. Prove that if ##X## is connected, a locally constant function on...

Hi, I'm trying to find the residue of $$f(z) = \frac{z^2}{(z^2 + a^2)^2}$$ Since I have 2 singularities which are double poles. I'm using this formula $$Res f(± ia) = \lim_{z\to\ \pm ia}(\frac{1}{(2-1)!} \frac{d}{dz}(\frac{(z \pm a)^2 z^2}{(z^2 + a^2)^2}) )$$ then, $$\lim_{z\to\ \pm ia}...

If ##\hat{T} = -\frac{\hbar}{2m}\frac{\mathrm{d^2} }{\mathrm{d} x^2}##, then the expectation value of the kinetic energy should be given as: $$\begin{align*} \left \langle T \right \rangle &= \int_{0}^{L} \sqrt{\frac{2}{L}} \sin{\left(\frac{\pi x}{L}\right)}...

is the magnetic field B→. a state function and exact differential? I argued that it's a state function, what do you guys think

This is a textbook problem: now for part a) no issue here, the range of the function is ##-1≤f(x)≤299## now for part b) i got ##x≥-1## but the textbook indicates the solution as ##x≥0## hmmmmm i think, that's not correct...

I would like to find a function such that for $$a(x) \rightarrow 1~\text{for}~(x \gg x_c)$$ $$a(x) \rightarrow f(x)~\text{for}~(x \ll x_c)$$ What could be the ##a(x)## ? I have tried some simple functions but could not figure it out. Maybe I am just blind to see the correct result.

We have Rayleigh's dissipation function, defined as ## \mathcal{F}=\frac{1}{2} \sum_{i}\left(k_{x} v_{i x}^{2}+k_{y} v_{i j}^{2}+k_{z} v_{i z}^{2}\right) ## Also we have transformation equations to generalized coordinates as ##\begin{aligned} \mathbf{r}_{1} &=\mathbf{r}_{1}\left(q_{1}, q_{2}...

Hi, I was trying to solve the attached problem which shows its solution as well. I cannot understand how and where they are getting the equations 3.69 and 3.69A from. Are they substituting the values of θ₁ and θ₂ into Expression 1 after performing the differentiation to get equations 3.70 and...

Hey everyone! I got stuck with one of my homework questions. I don't 100% understand the question, let alone how I should get started with the problem. The picture shows the whole problem, but I think I managed doing the a and b parts, just got stuck with c. How do I find the largest region in...

The vector equation is ## v(x)=(e^x cos(2x), e^x sin(2x), e^x) ## I know the arc-length formula is ## S=\int_a^b \|v(x)\| \,dx ## I found the derivative from a previous question dealing with this same function, but the when I plug it into the arc-length function I get an integral that I've...

Sketch by hand the function determined as f(x) = 1/28 (7√−16x^2 + 16x + 5 + 12) and then From the sketch, determine the domain and range of f in interval notation. Hint: Interpret f as part of a circle. You must include in your solutions the inputs and outputs you used to help you sketch the...

Hi, I cannot figure out how they got Table 2.1. For example, how come when x=1, F_X(x)=1/2? Could you please help me with it? Hi-resolution copy of the image: https://imagizer.imageshack.com/img923/2951/w9yTCQ.jpg

Hi, I have to find the real and imaginary parts and then using Cauchy Riemann calculate ##\frac{df}{dz}## First, ##\frac{df}{dz} = \frac{1}{(1+z)^2}## Then, ##f(z)= \frac{1}{1+z} = \frac{1}{1+ x +iy} => \frac{1+x}{(1+x)^2 +y^2} - \frac{-iy}{(1+x^2) + y^2}## thus, ##\frac{df}{dz} =...

Given a 1D heat equation on the entire real line, with initial condition ##u(x, 0) = f(x)##. The general solution to this is: $$ u(x, t) = \int \phi(x-y, t)f(y)dy $$ where ##\phi(x, t)## is the heat kernel. The integral looks a lot similar to using Green's function to solve differential...

While studying the solution to a integral problem I found online I ran across a special function I am unfamiliar with. The integral is $$ \int_0^{\infty}\frac{t^{\frac{m+1}{n}-1}}{1+t}dt=\mathcal{B}(\frac{m+1}{n},1-\frac{m+1}{n}) $$ This certainly isn't the normal beta function. What is it...

Let ##S_n## denote ##\{1,\ldots,n\}##, where ##n\in\mathbb{N}##. Recall that the ##\textrm{sgn}## function maps a permutation of ##S_n## to an element in ##\{1,0,-1\}##. We want to rework the definition of ##\textrm{sgn}## because it is not sufficient for some proofs about determinants. For...

Attempt at a solution: \begin{aligned}Z=\sum ^{N}_{r=0}C\left( N,r\right) e^{-\beta \left[ -NJ+2rJ\right] }\\ \Rightarrow Z=e^{\beta NJ}\sum ^{N}_{r=0}C\left( N,r\right) e^{-2\beta rJ}\end{aligned} Let ##e^{-2\beta J}=x##. Then ##e^{-2\beta rJ}=x^{r}##. \begin{aligned}\therefore Z=e^{\beta...

I have an experimantally obtained time series: n_test(t) with about 5500 data points. Now I assume that this n_test(t) should follow the following equation: n(t) = n_max - (n_max - n_start)*exp(-t/tau). How can I find the values for n_start, n_max and tau so as to find the best fit to the...

I would say that what this matrix does is rotate e.g. a vector by ##\pi/2## clockwise. Am I right? I would like to check my understanding.

GRAPH WITH VALUES: Sorry I have a small dilema, I don't know if this is a exponential or polynomial function. I'd think its exponential but it doesn't have same change of factors.

I live close to the (landing) flight path to a major airport. The city has been working with the airport and the FAA to deal with complaints about the noise the jets make as they come in for a landing. I recently attended a meeting where various solutions were discussed, such as changing the...

Suppose a program computes or reads-in a symbolic mathematical expression like ##2x^2 - xy + y##. What's an effective way to cause the program to implement the expression as a function (e.g. implement ## f(x,y) = 2x^2 - xy + y##) when the programmer doesn't know in advance what the expression...

Hi, I am 16 year old and I am very interested in Physics. This summer I solved Schrödinger equation using griffiths' introduction to quantum physics and other sources. I achieved to get an exact solution of the wave function but I would like to plot it in a programm in order to get the 3d...

I am trying to understand the relationship between Fourier conjugates in the spherical basis. Thus for two functions ##f(\vec{x}_3)## and ##\hat{f}(\vec{k}_3)##, where \begin{equation} \begin{split} \hat{f}(\vec{k}_3) &= \int_{\mathbb{R}^3} e^{-2 \pi i \vec{k}_3 \cdot \vec{x}_3} f(\vec{x}_3...

I would like to check my understanding of this problem. There are the following possibilities: a. Isolated points where the gradient is 0. b. The level curves of height 0 c. The level curves of height 1. d. The level curves of height -1. e. None of the above. I would choose a, c, d. Where...

Tried this, but not sure how am I supposed to square the whole equation and then square root it since this will inevitably give me imaginary values. Am I supposed to ignore the imaginary values? Also, how can I find out the phase in this case? Usually, it's taking the exponents but in this case...

Hello, I have a few questions and I'd appreciate if you can please help me. 1. If I want to say "for every ## i \in \Bbb N ## and ## 0 \leq j \leq i ## define ## A_{i,j} := i ## and ## B_{i,j} := i \cdot j ## ", then is the logical formula used for this is as such?: ## \forall i \in \Bbb N...

##A=(-1,1)## ##B=[0,1)## Define ##f:A\longrightarrow B## by ##f(x)=x^2## Set ##X=A##. ##f(X)=\{f(x):x\in X\}=\{x^2:x\in(-1,1)\}=B## ##f^{-1}(f(X))=\{x\in A:f(x)\in f(X)\}=\{x\in (-1,1):x^2\in B\}=X## Now choose a non-zero point ##y\in f(X)##. There are two pre-images of this point: ##x,-x\in...

ok I don't think MHB will process a newcommand but I don't know how to put this in the after \begin{tikzpicture} line the problem with posting pic here is eventually they get remove and OP is useless... this tikz code renders in overleaf but I also have many newcommands in preamble

Summary:: According to Yale’s University PHYS: 200: v*(dv/dt) = d(v^2/2)/dt Could someone explain how has he reached that conclusion? He claims to be some standard derivation rules, but I can’t find anything about it. As much as I can tell: (dv/dt)* v = v’ * v = a* v thanks! [Moderator's...

Last week I encountered a problem in my graduate project. The project was about designing an autonomous and mobile robot that picks up 9 glass tiles from a stack point and place them into a 3x3 matrix with minimum tolerance. I am using a DC motor with an infinite turn potentiometer for closed...

What is the difference between an absolutely continuously differentiable function and a wave? Are all absolutely continuously differentiable equations waves?

Please help me I am struggle with this question Thank you in advance

Can you please help me how to do it ? I am really struggle with this question. Thank you in advance

I came across it in the derivation of Gauss' law of electric flux from Coulomb's law. I did some research on it, but the wikipedia page about it was slightly confusing. All I know about it is that it models an instantaneous surge by a distribution. However I am still perplexed by this concept...

Can you help me with this two questions I am really struggle on how to do it Please help me Thank you in advance

I'm trying to see if I can calculate the peak draw weight of my bow based on the draw length and the velocity of the arrow and a known shape of a curve, but I'm not quite sure how to make such a function, if there even is such a way. This is the shape of the draw weight plotted against...

Hi All I am currently doing Master in data science. I came across the function PDF probability density function which is used to find cumulative probability(range) of a continuous random variable. The PDF probability density function is plotted against probability density in y-axis and...

My trial : I think ## \int ~ dy ~ e^{-2 \alpha(y)} ## dose not simply equal: ## - \frac{1}{2}e^{-2 \alpha(y)} ## cause ##\alpha## is a function in ##y ##. So any help about the right answer is appreciated!

I have calculated $Z$s as $$ \begin{aligned} Z_1 & = 1 + \frac{3g^2}{16\pi^2} \left[ 2 C(R) - \frac12 T(A) \right] \frac1{\epsilon} + \cdots, \\ Z_2 &= 1 + \frac{3g^2}{8\pi^2} C(R) \frac1{\epsilon} + \cdots, \\ Z_3 &= 1 + \frac{g^2}{24\pi^2} \left[ 5 T(A) - T(R) \right] \frac1{\epsilon} +...

Hello! Let's say we have a wave function. Maybe it's in a potential well, maybe not, I think it's arbitrary here. This wave function is one-dimensional for now to keep things simple. Then, we use a device, maybe a photon emitter and detector system where the photon crosses paths with the wave...

The ac signal is converted to DC signal which is connected to a capacitor to filter the DC signal. The filter DC signal is step down for 12volt to 5 volt using a voltage regulator. The regulated DC signal is connected to a crystal oscillator that converts the DC signal to a square wave signal...

Does anyone knows the name of the probability density function f(y) =aeby-cy2

I'm used to calculating Jacobians with several functions, so my only question would be how do I approach solving this one with only one function but three variables? I think our function becomes (s^2+sin(rt)-3)/since we are looking for J(f/s). So then would our Jacobian simply be J=[∂f/∂s...

Since z=0, the only variable that counts is x. So the solution would be: $$\frac {f \left(a + \Delta\ x, b \right) - f(a,b)} {\left( \Delta\ x\right)}$$

E.g. all real numbers could be a domain but not necessarily, etc. Am I right?

Hello, I don't know to solve this exercise: Let $\mathcal{B}_\mathbb{R}$ the $\sigma-algebra$ Borel in $\mathbb{R}$ and let $\mu : \mathcal{B}_\mathbb{R} \rightarrow{} \mathbb{R}_{+}$ a finite measure. For each $x \in \mathbb{R}$ define $$f_{\mu} := \mu((- \infty,x]) $$ Prove that: a)...

I followed a demonstration in one of my electromagnetism books, but it is not clear to me. My problem is at the starting point. The book begins by considering the office defined in the following way: $$Q=\int d^4xJ^\alpha(x)\partial_\alpha\theta(\eta_\beta x^\beta)$$ where...

hi guys I was trying to verify the integral representation of incomplete gamma function in terms of Bessel function, which is represented by $$\gamma(a,x) = x^{\frac{a}{2}}\;\int_{0}^{∞}e^{-t}t^{\frac{a}{2}-1}J_{a}(2\sqrt{xt})dt\;\;a>0$$ i was thinking about taking substitutions in order to...

I have no problem in following the literature on this, i find it pretty easy. My concern is on the derived function, i think the textbook is wrong, it ought to be, ##S^{'}(t)##=##\frac {4t} {\sqrt{1+4t^2}}=0## is this correct? if so then i guess i have to look for a different textbook to use...