Domain Definition and 743 Threads

Prove that in a prinicpal ideal domain, two ideals (a) and (b) are comaximal if and only if a greatest common divisor of a and b (in which case (a) and (b) are said to be coprine or realtively prime) Note: (1) Two ideals A and B of the ring R are said to be comaximal if A + B = R...

Homework Statement Prove Z[x]/pZ[x] is an integral domain where p is a prime natural number. Homework Equations I've seen in notes that this quotient ring can be isomorphic to (z/p)[x] and this is an integral domain but I don't know how to prove there is an isomorphism between them and...

When I plot y = 3x^{\frac{4}{3}}-\frac{3}{32}x^{\frac{2}{3}}, I get: https://dl.dropbox.com/u/5653705/pfgraph1.png but I don't see any reason for the domain to be restricted to x > 0 . There is only a cubic root, which is well defined for negative numbers ... I've tried on a few...

Hi, I'm studying Calculus 2 now, and I am a litle bit confused in this question. 1- Determine and plot the domain of the function of two variables a) f(x,y)=\sqrt[]{1+x^{2}+y^{2}} x^{2}+y^{2}\geq -1 doing x^{2}+y^{2}= -1 i guess this is the graph of Hyperbole, but my teacher said is a...

what's negative frequency? can we say s=-3 in example bode plot ! or we can say it's unstable at w=-3 rad/s that's I mean is there negative frequency??

Is something wrong in my assertions below? Suppose we have two quantum systems N and X. Let N is described by discrete observable \hat{n} (bounded s.a. operator with discrete infinite spectrum) with eigenvectors |n\rangle. Let X is described by continuous observable \hat{x} (unbounded s.a...

Homework Statement x is square free (not divisible by a perfect square) on Z Homework Equations Z meaning all integers. The Attempt at a Solution I did a similar problem earlier that asked for the expression for prime numbers on the Natural numbers domain. For that problem the...

Suppose that we have rigged Gilbert space Ω\subsetH\subsetΩ\times (H is infinite-dimensional and separable). Is the Ω a separable space? Is the Ω\times a separable space? Consider the complete set of commuting observables (CSCO) which contain both bounded and unbounded operators...

i am currently reading griffith's book on electrodynamics, though he does an excellent job with the theory (along with all the sloppiness in math) he does not really answer the question, how well does this theory apply to the real world. i know Newton's theory is valid at v<<c, however in...

What is the domain of sqrt(2x-x^3) I thought this would be pretty straight forward but I am completely stumped. So obviously square root is not defined for (-inf, 0), therefore I said: 2x-x^3 >= 0 Working this out you get x <= sqrt(2) as the domain. That is partially correct but I can't for...

I am going through a past paper for an upcoming exam and I want to check that I am approaching this question correctly. H(s) = 1/(s2 + 8485.28s + 36x10-6) Calculate and plot frequency response for 0 rad/s, 500 rad/s, 1000 rad/s and 10000 rad/s. I am under the impression that the 's'...

On one side one can define second-order variables as ranging over all elements of Pk(M) for all natural numbers k (P=power set of M, M is the universe of the model, superscript being iteration). On the other side it is sometimes defined as ranging over all first-order relations and predicates...

Homework Statement If we have R = \{ \frac{1}{2}(a+b\sqrt{2}) \}. Homework Equations The Attempt at a Solution For the identity, we have a=2 and b=0. So the identity is not equal to zero. Also, there can't be any zero divisors, because...

So just had this question as extra credit on a final: Let D be an integral domain, and suppose f is a non-constant map from D to the non-negative integers, with f(xy) = f(x)f(y). Show that if a has an inverse in D, f(a) = 1. Couldn't figure it out in time. I was thinking the way to go...

if a function f is differentiable of [0,2pi] can I integrate its derivative df on [-pi, pi]?

Homework Statement Find the Fourier transform of the following signal JUST by using the FT table and the FT properties x(t) = sin(t) -pi<=t<=pi 0 otherwise NOTE: I am using CONVOLVED WITH as a substitute for * (the real convolution operator) because I cannot express...

Hey guys, Does anyone know of a good, free domain coloring grapher? I can't find a decent one anywhere.

Let R be an integral domain. Then R is a PID if the following 2 conditions hold: 1) any 2 elements a, b \in R have a greatest common divisor which can be written as ra + sb for some r, s \in R . 2) If a_1, a_2, ... are nonzero elements of R such that a_{i+1}|a_i for all i, then there is...

I was wondering if there is an easier way to solve circuits using matrix inversion if I have complex numbers. So far I've been doing them by hand with 2x2 matrices. It isn't hard, but it takes a while. I have a test coming up in about a week which will have 10 questions that have to be solved in...

Check this out: \int_{-1}^{0} ln(x) dx u=ln(x), dv=dx du=\frac{1}{x},v=x \int_{-1}^{0} ln(x) dx= x\space ln(x) \|_{-1}^{0} -\int_{-1}^{0} \frac{x}{x} dx =x\space ln(x) \|_{-1}^{0}-x \|_{-1}^{0} =\lim_{x\to0} x\space ln(x)-(-ln(-1))-(0-(-1)) \lim_{x\to0} x\space ln(x)=lim_{x\to0}...

I'm having an interpretation problem with the notation used in physics, under the integration sign. What is the proper interpretation of the domain of integration symbol, on the integration sign ? To be more precise, consider a function F(x) of one or several variables. Its integral on a...

Homework Statement when I'm trying to calculate and show the graphic of the phase characteristic i don't understand why the domain range of the arctan function is [0,90] : Homework Equations The Attempt at a Solution shouldn't be the domain of arctan function between...

Find the domain of the function f(x)= (3x+5)1/2. Not quite sure how to even start this...

Homework Statement Consider some simple single-variable function such as f(x)=2x+1 The domain,codomain and range of the above function is set of all real numbers. But why not the set of all complex numbers? if x = 3+4i f(x) = 6+8i + 1 = 7+8i,so it is possible for x to be a complex no...

Homework Statement Hi All, I'm just trying to practice graphing signals in frequency domain and I came across a stiuation I wasn't familiar with. If the exp() has a constant*t in it I'm not sure how to graph it, I remember that just cos it like a double sided exp(jwt) but with half the...

Homework Statement Determine the domain and range of the function y=5−e^(−x/2). I am having some trouble with this. Does any have some tips to get started, or a good link to a instructional video? My textbook does not have this section :( Thanks all! Homework Equations The...

Hi All, I'm just trying to practice graphing signals in frequency domain and I came across a stiuation I wasn't familiar with. If the exp() has a constant*t in it I'm not sure how to graph it, I remember that just cos it like a double sided exp(jwt) but with half the magnitude. I've attached a...

Homework Statement What is the domain of f(x,y)=x^y The Attempt at a Solution I thought it would be all real pairs except (0,0) but it is x>0 and y all real numbers?

Homework Statement How to solve part (iv) & (v)Homework Equations general form : y = a(x-h)^2 + kThe Attempt at a Solution In part (iv) for finding domain and range I converted g(x) in general form and then compared it with general form.

Hi all, This question is on the Hilbert transform, particularly on the domain of the input and output functions of the Hilbert transform. Before rising the question, consider the Fourier transform. The input is f(t) and the output is F(\omega). The function f and F are defined over...

Homework Statement If the maximum allowable width, w, of the wall is 100 cm, what are the domain and rand of the function? Remember to include units for both the domain and range. Homework Equations w(d)=2d w(d)=100, so 100=2d, there fore d=100/2 The Attempt at a Solution...

I have a question about domain and range.What are the domain and range of cos(e^{-×}) and \frac{\left|2×-1\right|}{sin(\frac{1}{2}\pi×-\pi)} ?? Thanks..

What must actually be specified in order for a function to be fully defined / or in what combinations if not all 3 need to be specified? I.e - from knowing the function you can determine the co-domain - e.g - if it is specified that real functions are going in, and for something simple like 2x...

I am reviewing for the mathematics GRE subject exam, and I have this review book, and in the book when they speak of the characteristic of a ring they give the following def "Let R be a ring. The smallest positive integer n such that na = 0 for every a in R is called the characteristic of the...

The "correct" domain of self-adjointness for the Laplacian Consider the Hilbert space L^2(\mathbb R^d), and consider the Laplacian operator \Delta on this space. We want to find a domain, D(\Delta) \subset L^2(\mathbb R^d), such that this guy is a self-adjoint operator. We have been talking...

This problem seems a little overwhelming at the point. I am not sure on where and how to start. Suppose that a uniform thermal gradient in the +x direction exists in a very large (i.e. effectively infinite) domain of conductivity $k_2$ such that the temperature field $u_{\infty}(r,\theta)$ can...

Homework Statement I have this function of two variables: f(x,y)=\sqrt{4xy-3y^2} Where I have to declare the domain of the function. Homework Equations Since you cannot take the square root of something negative, when dealing with real numbers. The square root has to be zero or...

Homework Statement Find the derivatives at an arbitrary point x in the domain of the following functions f_i: D_i → ℝ, where for 1 ≤ i ≤ 6 the domain D_i is the maximal subset of ℝ on which the mapping is defined - you don't have to determine the domains. Homework Equations a) f_1 (a) =...

Homework Statement I need to find the domain of sin(arcsin(x)). Now I "know" that this is xε(-1,1), but I don't understand why. If the range of arcsin(x) is (-∏/2,∏/2), then shouldn't this be the domain of sin(arcsin(x))? Help appreciated.

Homework Statement This question here. Can someone please tell me why we have 2 cases to consider. http://archives.math.utk.edu/visual.calculus/0/domain.1/8.html I don't understand why it says we have 2 cases to consider. 8 x - 4 > 0 and x - 3 > 0. 8 x - 4 < 0 and x - 3 < 0...

I need help understanding how frequencies are distributed when taking the FFT of an Image and how can i determine those frequencies, I really need a detailed and easy tutorial with practical example to understand it. When we take the FFT of a Sine/Cos wave we can easily see its frequencies in...

Homework Statement Find domain and range of f(x,y)=\frac{-3y}{x^{2}+y^{2}+1} Homework Equations The Attempt at a Solution It's quite obvious the the domain is R^{2}, because there are no possible values of x or y that would make the denominator 0. But as for the range, my...

As I understand it, there are the terms: domain - variables that are input to a function range - the variable that is output to a function This works well in a canonical single equation system like z = f( x , y ), but breaks down in an implicit function or set of equations. I was...

Hello guys, I need some assistance in calculating the domain of this function: f(x)=ln(sin(pi/x)) I started with sin(pi/x)>0 due to the ln function. From here 0<(pi/x)<pi. That lead me to some calculations giving x>1, but obviously I have periods of 2*pi to include. The answer is: x>1...

I have came up with a solution for this - in order for this function to be defined, we must have an x and y between negative infinity up to and including the number 1. If asked to graph this domain, does the domain lie on the x-y plane of three dimensional space, and is it the intersection of...

Homework Statement Find the domain and range of the following function without the use of a calculator: f(x) = sec (pi x/4) Homework Equations As far as I know, this problem doesn't specifically require "equations". Therefore I am leaving this section blank. Not because I am a...

Homework Statement Ok, I just worked out a composite function, and it left me with: √2x^2+5) Now, how do I find the domain from that? I don't understand that my text says the domain of that is just all Real numbers ? What makes this different than other square functions that we are...

Homework Statement Problem: http://archives.math.utk.edu/visual.calculus/0/domain.1/f6.gif Homework Equations The Attempt at a Solution Look at the pic, and tell me again why I can't do this: http://i47.tinypic.com/bhadsj.jpg

Homework Statement If f(x)=√x and g(x)=√(x^2+1) what is the domain of g(f(x))? The Attempt at a Solution I cam up with (-∞,+∞) since the square root of a square is always positive, am I missing something here?

Homework Statement let A be a UFD and K its field of fractions. and f\in A[x] where f(x)=x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0} is a monic polynomial. Prove that if f has a root \alpha=\frac{c}{d}\in K,K=Frac(A) then in fact \alpha\in A I need some guidance with the proof. Proof...