Domain Definition and 743 Threads

Hi all, I'm trying to see if my question/logic makes sense. Suppose I have a classical Ising model on a 2-D Square lattice with ##N## sites and 0 external field. There is an exact formula for the average internal energy per site, and at criticality it turns out to be ##u = -\sqrt{2}## where I...

The ####x partial derivative is equal to $$L \frac{4x}{5(x^{2}+y^{2})^{\frac{-3}{5}}}$$ and the partial for ##y## is $$L \frac{4y}{5(x^{2}+y^{2})^{\frac{-3}{5}}}$$ Using the limit definition of partial derivatives I got the partial wrt ##x## is $$L \frac{h^{\frac{4}{5}}}{h}$$ which doesn’t exist...

##\small{\texttt{(I could solve the for the upper limit explicitly.}}## ##\small{\texttt{However, not the same for the lower limit, except via inspection.)}}## I copy and paste the the problem as it appeared in the text. ##\rm(I)## : ##\texttt{The domain :}## The domain of the function is...

Attempt : Let me copy and paste the problem as it appeared in the text. Please note that the given problem appears in part (b), which I have underlined in red ink in this way ##\color{red}{\rule{50pt}{1pt}}## Clearly the domain is ##\boxed{\mathscr{D}\{f(x)\}...

Am refreshing on this, For the domain my approach is as follows, ##(f-3g)x = f(x)-3g(x)## ##=x-3-3\sqrt{x}##. The domain of ##f-3g## is given by ##f∩g = [{x: x ≥0}]## We have ##y= x-3-3\sqrt{x}=(\sqrt x-\frac{3}{2})^2-\dfrac{21}{4}##. The least value is given by...

I don't know if it's a turnkey operation or if it's just the domain name itself, but thescienceforum.com is up for auction. Current bid $493...

Let ##D## be a smooth, bounded domain in ##\mathbb{R}^n## and ##f : D \to (0, \infty)## a continuous function. Prove that there exists no ##C^2##-solution ##u## of the nonlinear elliptic problem ##\Delta u^2 = f## in ##D##, ##u = 0## on ##\partial D##.

Hello All, I am somewhat familiar with FFT and iFFT and its uses. However I have an issue when I edit in Freq domain and try to get back to time domain . I have an audio signal in time domain that I transform to frequency domain using an FFT routine in block sizes of N points. (in my case 256...

I am studying this now...i just want to check that my reasoning is fine,... ##[1+\sqrt{-3}]## can be factorised to ##[2⋅-1]## and ##-1## is a unit and ##2## is irreducible, therefore ##\mathbb{z}[\sqrt 3]## is a UFD.

What is the limit of the function as x goes to -5 (e.g. in the graph below) if the domain of the function is only defined on the closed interval [-5,5]? I realize that the right hand limit DOES exist and is equal to 3, but the left hand limit does not exist? So does that mean that the overall...

For this problem, The solution is, However, when I tried finding the domain myself: ## { x | x - 1 ≥ \sqrt{5}} ## (Sorry, for some reason the brackets are not here) ##{ x | x - 1 ≥ -\sqrt{5}} ## and ## { x | x - 1 ≥ \sqrt{5}}## ##{x | x ≥ 1 -\sqrt{5} }## and ## { x | x ≥ \sqrt{5} + 1}##...

For this problem, The solution is, However, I tried solving this problem by using the definition of composite function ##f(g(x)) = f(\frac{4}{3x -2}) = \frac{5}{\frac{4}{3x - 2} - 1} = \frac{5}{\frac{6 - 3x}{3x - 2}} = \frac {15x - 10}{6 - 3x}## which only gives a domain ##x ≠ 2##. Would some...

Hi everyone I have the solution to this question, but I'm not sure I understand it. Why is the domain of the composite function and not [0, pi]? Is it because tan^-1 (0 and R+) will always give a value between (-pi/2, pi/2)? I.e. the domain of the composite function refers to x. Is 0...

Problem statement : Let me copy and paste the problem as it appears in the text : Attempt 1 (from text) : The book and me independently could solve this problem. I copy and paste the solution from the book below. Attempt 2 (my own) : The problem should afford a solution using the second idea...

I was wondering if anyone could walk me though a better explanation on how to get the given results for these two questions. The solutions posted by my professor aren't really clear to me so if anyone is able to better explain how to get the solution it would be much appreciated!

Hello. Considering this DE; $$ x^7 x' = (x^8-300)t^6 $$ with inital value x(0) = -2 Now the solution for the initial value should be C = -44; And for x(t) I get ; $$x(t) = (-44 e^{\frac{8}{7} t^7} + 300)^{\frac{1}{8}}$$ Now to get the biggest domain of definition I did this; $$ -44...

I found that the a) invariant points are all points on y-axis b) invariant lines are y-axis and ##y=c## where ##c## is real I am confused what the final answer should be. How to state the answer as "subset of domain"? Is it: $$\{x,y \in \mathbb R^2 | (0, y) , x = 0, y=c\}$$ Thanks

Is it possible for Python matplotlib to plot in one graph a domain dependent function? For example, suppose there is a function where y=x from 0< x <=5, y = x sq when 5<x<=7 and y=2x+9 when x>7. Is it possible in Python to plot this with one plot on one graph? If so, how would it be done?

$$n_\vec{k} = \omega a^2(\vec{k})\tag{1}$$ One way is to write the inverse Fourier transforms of the terms above. So, eqn (1) becomes $$\int\mathrm{d}^3x\ n(\vec{x})e^{-i\vec{k}\cdot\vec{x}} = \omega \int\mathrm{d}^3x^\prime\ a(\vec{x^\prime})e^{-i\vec{k}\cdot\vec{x^\prime}}...

At the risk of waiting hours on simulations of a sensor, I was wondering if I could use infinite element domain on COMSOL to simplify it. The first image consists of what I would like to simulate but found out that the simulation time is a huge factor as I have a lot to simulations to conduct...

The time domain analysis is easier to plot compared to analyzing the frequency with respect to the phase. But, LTSpice makes it look really easy. So, for a small signal AC analysis, LTSpice does use a AC voltage source for its frequency domain analysis function. This must be a really convenient...

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...

Hi PF I have a quote from Spanish 6th edition of "Calculus", by Robert A. Adams, and some queries. I translate it this way:"The inverse of secondary trigonometric functions can easily be calculated with the reciprocal function. For example DEFINITION 13 The inverse function of secant ##sec^{-1}...

Hello folks, I want to simulate a 2D heat transfer process in the subsurface on a region which is infinite on the r-direction. So, as you know, the very basic way to model this is to draw a geometry that is very long in the r direction. I have done this, and the results that I obtain is correct...

$\tiny{GRE.al.4}$ Find the domain of $f(x)=\sqrt{x^2-25}$ a. $[x\le-5]U[x\ge 5]$ b. x=5 c. $5 \le x$ d. $x\ne 5$ e. $\textit{all reals}$ well just by observation because of the radical I chose c. but was wondering if imaginary numbers could be part of the domain alto it is not asked for here...

$\tiny{\textit{5.6.28}}$ Find $x \quad |2x-5|<9$ divide into 2 solution sets $\begin{array}{rl|rl} (2x-5)&=9&-(2x-5)&=9 \\ 2x&=14&-2x&=4 \\ x&=7&x&=-2 \end{array}$ x is $-2<x<7$ kinda maybe...

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

$\tiny{s5.t.5 Kiaser HS}$ Find domain asymptotes. $g(x)=\dfrac{2x^2-14x+24}{x^2+6x-40}$ $\begin{array}{rll} \textsf{factor}&=\dfrac{2(x-3)(x-4)}{(x-4)(x+10)} =\dfrac{2(x-3)\cancel{(x-4)}}{\cancel{(x-4)}(x+10)} =\dfrac{2(x-3)}{x+10}\\...

Ron Larson stated: "The domain of a function can be described explicitly or it can be implied by the expression used to define the function. The implied domain is the set of all real numbers for which the expression is defined." 1. How is a function defined explicitly? 2. How is a function...

I want to simulate the time domain data for a rotating radar. I assume that the space around the radar is filled up with a very big extended object and it moves with a constant speed in one direction. Picture attached.I don't take range information here. I am only concerned about the velocity as...

I am trying to simulate and process the Doppler signals. My main problem is a little more complex so I am only posting a simple version of it. Task1: I have a time-domain signal with the velocity of the target as mu. I need to change the velocity to mu cos(theta) where theta is a vector from 0...

The problem I am having is simple. I have a Gaussian spectrum initially. Like this, Process 1: S = m0/sqrt(2*pi*sigma^2) * exp(-(vel_axis - mu).^2/(2*sigma^2)); Here, mu is the mean velocity (frequency) and sigma is the standard deviation. vel_axis is the axis on which I am calculating this...

I am having some trouble find the domain with this function: ##f(x)=\frac{1}{\sqrt{x^2-4x\cos(\theta)+4}}## and ##\theta\in[0,\pi]##.I know that the denominator needs to be greater than 0. So ##\sqrt{x^2-4x\cos(\theta)+4}>0##. I squared both side of the inequality, ##x^2-4x\cos(\theta)+4>0##...

Dear Everyone, I am having some trouble find the domain with this function: $f(x)=\frac{1}{\sqrt{x^2-2x\cos(\theta)+4}}$ and $\theta\in[0,\pi]$. My attempt: I know that the denominator needs to be greater than 0. So $\sqrt{x^2-2x\cos(\theta)+2}>0$. I squared both side of the inequality...

Say we have a transform of a line profile that extends out to the Nyquist frequency such that you cannot see the noise level, what could you change in your spectrograph arrangement that would allow you to see the noise level in the Fourier domain? My thought is that we can apply a filter, P(s)...

For n=1 (ln(ln(x))), the domain is the set of all real x>1 ; for n=2, the domain is the set of all real x>e ; for n=3, the domain is the set of all real x>e^e ; for n=4, the domain is the set of all real x>e^(e^e) ...Thus, for a general n the domain is the set of all real x greater than...

Hi, I am interested in understanding the relationships between Fourier series and Fourier transform better. My goal is 1) Start with a set of ordered numbers representing Fourier coefficients. I chose to create 70 coefficients and set the first 30 to the value 1 and the remaining to zero. 2)...

y = 2sin(x) -1≤ sin(x) ≤ 1 -2 ≤ 2sin(x) ≤ 2 so -2 and 2 are the max/min limits but the domain is -π < x ≤ π Do I find the values of x that outputs -2 and 2 and show that they are within the domain ?

Convolving two signals, g and h, of lengths X and Y respectively, results in a signal with length X+Y-1. But through convolution theorem, g*h = F^{-1}{ F{g} F{h} }, where F and F^{-1} is the Fourier transform and its inverse, respectively. The Fourier transform is unitary, so the output signal...

the solution

I'm quickly reading at https://boingboing.net/2019/08/01/80pct-pd.html?fbclid=IwAR3pMnu6G0d-M6Ldf_i5muga3g_m0NVkMQ5u6NiE-f_FapGKftJO76_hxbw that about 80% of books published in the US between 1924 and 1963 are in the Public Domain now. Textbooks are known to be very expensive, even when they are...

Good day as said in the title i need the domain of definition of of the function f(x,y)=x^y for me as x^y=expontial (y*ln(x)) so x>0 but the solution said more than that I really don't understand why we consider the case (0,y) in which while should be different from 0, because I will never...

In August, "Quantum Information Processing" published an article describing a full FFT in the quantum domain - a so-called QFFT, not to be confused with the simpler QFT. According to the publication:

Are topological defects such as domain walls really discontinuous (implying infinite slopes of the fields) or only approximately ?

##f(x)=-x^2 + 3## so ##f^{-1} (x)=- \sqrt{3-x} ~, x \leq 3## ##ff^{-1} = - (- \sqrt{3-x})^2 + 3 = x## and the domain will be ##x \leq 3## ##f^{-1} f = - \sqrt{3-(-x^2+3)} = -x ## and the domain will be ##x \leq 0## My question: ##ff^{-1} (x)## or ##f^{-1} f(x) ## is not always equal to...

The domain and range of this function will be the same. We can let ##𝑓(𝑥)=\sqrt{x},𝑥≥0## However, ##𝑦=𝑓(𝑥)≥0##, so the domain and range of ##f## are ##[0,+∞)## And since ##f## is a function, ##f^{-1}s## domain is the range of ##f## and ##f^{-1}s## range is ##f’s## domain. In other words...

This is a problem about converting from the phasor to time domain. I am having trouble following the math that the textbook is doing. I was thinking the final answer should be: i(d, t) = 0.20 cos(ωt + βd + 159◦) - 0.091 cos(ωt − βd + 185.6◦) emphasize the minus 0.091 instead of plus 0.091 as...

i have to find such domain z=x+iy , y,x∈ℝ , |y-x|<=2, |x|<=2 i'm confused with |y-x|<=2, how should i proceed ? with abs of x i am ok.

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

Solve for y: $\quad |y+3|\le 4$ a.$\quad y \le 1$ b.$\quad y\ge 7$ c.$\quad -7\le y\le1$ d. $\quad -1\le y\le7$ e. $\quad -7\ge y \ge 1$ Ok I think this could be solved by observation but is risky to do so...