domains Definition and 114 Threads

Homework Statement Question 1: Which of the following define y as a function of x on R (Real number). Explain for each why they are/ are not function. a) 4x^3 + y = 6 b) x - y - square root x = 8 c) x = cos^2 y d) y = (2x + 3) / (x - 1) Question 2: Let g(x) = sin(x) and h(x) = 1/x...

Most biology textbooks state that life can be classified into three domains: bacteria, eukarotes, and archaea. This classification began from early studies looking at the evolutionary relationship between these three groups of organisms that concluded that all archaea are more similar to...

In short, what I am asking is what is the reason/cause of the formation of ferro- magnetic/electric domains, and how would one calculate the 'driving energy' in order to predict domain formation? If we consider a dipole system that experience “ferromagnetic” coupling (my question does not...

First of all, I would like to appologize if I'm posting this in the wrong section, although I believe it to fit this area. I'm usually not convinced when books claim to have found the natural domain of certain functions. For instance, this book I've been reading has defined the natural domain...

In this video (from 27.00 - 50.00, which you don't need to watch!) a guy shows how you can solve the general second order ode y'' + P(x)y = 0 using perturbation theory. However he points out that the domain must be finite in order for this to work, I'm wondering how you would phrase a question...

Just a further (very basic!) question: Is the following argument - working from definitions - correct Does (a) + (b) = (a,b)? --------------------------------------------------------------------------------- By definition (Dummit and Foote page 251) (a, b) = \{r_1a + r_2b \ | \ r_1...

Hi there. I am currently taking "College Math 1" at the local CC and I have encountered something that confuses me regarding rational expressions and their domains. The definition given by the textbook for rational expressions is: "the set of real numbers for which an algebraic expression is...

I am reading Dummit and Foote Sections 9.3 Polynomial Rings that are UFDs. I have a problem understanding what D&F say regarding GCDs on page 306 at the end of Section 9.3 (see attached) D&F write: ====================================================================================== "we...

Dummit and Foote, Section 8.2 (Principal Ideal Domains (PIDs) ) - Exercise 4, page 282. Let R be an integral domain. Prove that if the following two conditions hold then R is a Principal Ideal Domain: (i) any two non-zero elements a and b in R have a greatest common divisor which can be...

Hi all, first post :) I have a system of z-propagated nonlinear PDEs that I solve numerically via a pseudo-spectral method which incorporates adaptive step size control using a Runge-Kutta-Fehlberg technique. This approach is fine over short propagation lengths but computation times don't...

In Dummit and Foote, Section 8.3 on Unique Factorization Domains, Proposition 10 reads as follows: Proposition 10: In an integral domain a prime element is always irreducible. The proof reads as follows: =========================================================== Suppose (p) is a non-zero...

Suppose we have the double integral of a function f(x,y) with domain of integration being some rectangular region in the 1st quadrant: 0≤a≤x≤b, 0≤c≤y≤d. Would the following transformation generally be acceptable? (I've quickly tried it out several times with arbitrary integrands and domains...

Homework Statement Consider the ring Z/mZ, show that S = {[0], [a], [2a], · · · , [m − a]} forms a (possibly nonunitary) subring of Z/mZ when a divides m. (i.e. show that (S,+, ·) is closed the usual addition and multiplication. (We are not require to find a multiplicative identity)...

When I was working on a rather difficult real-life math problem, I nearly found the solution. What I came up with was two inequalities: ##X≥\frac{2b-2}{2a+1}-1## and ##Y≥\frac{2b}{2a+1}-2## and the fact that ##X>Y##. However, ##X## must be an even integer and ##Y## must be an odd integer. Is...

While my math struggles continue. I find my self asking if this is the right major I want to chose (astrophysics) I'm in precalculus college level lol.. We are doing domains of comp functions.. and I find it all pointless.. I am very good at algebra and trig... Is this fog (x) stuff really...

Given is the function of Set V towards Set W where A is a subset of V and B is a subset of W. Questions: Does the range of the complement of A equal the complement of the range of A? Does the domain of the complement of B equal the complement of the domain of B?I am not entirely sure how to...

Let A be an integral domain. If c ε A, let h: A[x] → A[x] be defined by h(a(x))=a(cx). Prove that h is an automorphism iff c is invertible. This one really had me stumped. I have a general idea of what the function is doing. Now, assuming that h is an automorphism, we want to show that...

mod(x-6) > mod(x^2 - 5x + 9) Can anyone tell me about domain fixtures with mods using the above inequation? A real beginner, so have mercy, be elementry .:D

Dear Folks: Suppose \Gamma is a discrete subgroup of SL2(R), which acts on the upper half complex plane as Mobius transformation. F is its fundamental domain. If z is a vertex of F which does not lie on the extended real line ( that is R\bigcup\infty ) ,then must x be an elliptic point...

Hi PF , I am making a review article which is mainly based on low temperature physics , upon going through my search I have stumbled across the famous " Lambda transition" of super liquid helium. Paraphrasing what some of the books said : " In the He II domain a percentage of atoms in same...

Folks, When we are evaluating integrals like the following, what are we evaluating in terms of units etc. For example if I integrate Fdx I get an area which represents the energy where F is the force and d is the displacement so the units are Nm etc. 1) Integrals over intervals ...

Homework Statement http://imageshack.us/photo/my-images/15/unledflsq.png/Homework Equations A simply connected domain D in the complex plane is an open and path connected set such that every simple closed path in D encloses only points of D.The Attempt at a Solution The answers are a,c and d.I...

Homework Statement Let A = \{(x, y, z) \in \mathbb{R}^n : 0 \lt x \leq 1, 0 \lt y \leq 1 - x^2, 0 \lt z \leq x^2 + y\}. Define f : A \rightarrow \mathbb{R} by f(x, y, z) = y for each (x, y, z) \in A. Accept that Fubini's theorem is applicable here. Find \int_A f. Homework Equations Fubini's...

Hey guys, I'm doing some multivariable calculus atm, and I need some help with the Domains of some multivariable functions... 1) f(x,y) = 3x^2 + 2y The problem I'm having here is I basically forget the definition of domain... would it be for all x and y even though there are two whole quadrants...

I have some understanding of how to solve problems involving compact domains. Set the gradient to zero and solve for x and y, and then try to parameterize if needed to find max/min over the border of the domain. The thing is, my book doesn't go into much detail on how to do optimize functions...

Homework Statement Ok, this is not really a problem, but I need help on understanding the basics of sin, cos, tan, and their inverses. i was looking at http://www.analyzemath.com/DomainRange/domain_range_functions.html and it was saying that the domain for sin and cos is (-inf , + inf)...

Homework Statement How do you find the domain of y=ln(6-x) ? The Attempt at a Solution do i have to set 6-x greater than or equal to 0 to then get that x is less than or equal to 6, or is there more to it than that? I'm confused on where the ln part comes in. Please assist.

If you have an equation with a variable which isn't defined for a given value or values, how do you express this in proper notation? For example: x=1/((y-2)(y-3)) Do I write simply " y<2 or 2<y<3 or 3<y" or is there a better way to express it? Thx

Supposing I have a function f(x). Let us suppose that f-1(x) has the same equation as f(x). Will the domain and range as defined for f(x) be the same as for the inverse ?

Im preparing for a CLEP test in precalculus. As part of my prep, I need to review identifying domains of functions. I have a question about writing domains in standard notation. I was hoping someone could explain a bit the style. For an example: x-2 / x^2 -2x -35 As a rational...

Homework Statement Consider the integral domain a + b \sqrt{10} . Show that every element can be factored into a product of irreducibles, but this factorization need not be unique. Homework Equations The Attempt at a Solution I know that this is not a unique factorization...

I'm having a bit of trouble with some ring theory I've been reading about, specifically unique factorization domains. I'm not really clear on how one would go about showing that an element can be factored into irreducibles Homework Statement Let R be an integral domain such that every prime...

Is it always true that the domain of f(g(x)) is the intersection of the domains of f(x) and g(x)? I've been having trouble with this and this answer would make me fully understand this concept. Thanks to everyone!

Homework Statement Given an example of two different functions f and g, both of which have the set of real numbers as their domain, such that f(x)=g(x) for every rational number. 2. The attempt at a solution I have yet to figure a way to approach this problem. Since it appears as...

Popularized treatments of quantum mechanics describe it as applicable to the behavior of submicroscopic particles, while relativity applies to the very large (i.e., astronomical). This seems totally arbitrary to me. Where is the boundary between the 2 domains? Atoms? Neutrons? Quarks? Or is it...

Homework Statement The forward-back function is f (t) = 2t for 0\leq{t}\leq{3} , f(t)= 12-2t for 3\leq{t}\leq{6}. Graph f(f(t)) and find its four-part formula. First try t = 1.5 and 3. The Attempt at a Solution There are four possible composite functions from the two given...

Homework Statement 3. (a) Let f(x) = ln(x^2-1), and [itex]g(x)=\frac{x}{\sqrt{2-x}}[/tex] (i) Find the natural domains of f, g, f + g, \frac{f}{g}, and \frac{g}{f} Homework Equations N/A The Attempt at a Solution I know that the natural domain of f(x) is x belongs to real...

i'm working through the following text and I think I found an error please let me know if I'm totally wrong. Janusz, Gerald J. Algebraic Number Fields and I'm starting with the 3rd exercise on page 3. It is as following: let R be an integral domain and p a prime ideal of R. Show there...

Homework Statement For the following mappings, state the domain and the codomain. Determine whether the mapping is linear by using the definition of linearity: either prove it is linear or give a counterexample to show why it cannot be linear. i.) f(x1,x2,x3)=(2x2, x1−x3) ii.) g(x1, x2) =...

Hi there Suppose I have data of the format {x,y+i z} where x,y,z is real and i is the imaginary unit. I'm trying to make a FindFit of some nasty model that, suppose for simplification is f(x) = a^b*x^2+exp(a)*b*i*x (domain is real, codomain is complex and a,b are Real) and can be written...

Please Help! Mathematica ignoring variable domains I had to calculate an integral, which involves real as well complex parts. As mathematica takes all variables to be complex by default I used the elements function to define that certain variables were Reals. But it doesn't change the...

Homework Statement Let G be a finite group and let p >= 3 be a prime such that p | |G|. Prove that the group ring ZpG is not a domain. Hint: Think about the value of (g − 1)p in ZpG where g in G and where 1 = e in G is the identity element of G. The Attempt at a Solution G is a...

Homework Statement a) show that Q(√5i) = {r +s√5i | r,s in Q} is a subfield of C. b)show that Z(√5i) = {n + m√5i | n,m in Z} is a subring of C and find the units. The Attempt at a Solution a) Let a = r + s√5i, b = r - s√5i for a,b in Q(√5i). a + b = 2r, ab = r^2 + 5s^2, and -a= -r -...

what's a phasor? What's the phasor domain? I've worked with them in my courses and I can move from the time domain to the phasor domain, but I still don't quite intuitively get what a phasor is. In physics, we move between the frequency domain and time domain easily, but they're both...

I have a conceptual question. I have two physics books that say "In a common iron nail, the domains are randomly oriented. When a strong magnet is brought nearby, two effects take place. One is a growth in size of domains that are oriented in the direction of the magnetic field. This growth...

Homework Statement r(t)- ln|t-1| i , e^t j , sqrt(t) k find the natural domains. this is a problem as an example in the book. Homework Equations It gives an answer of (-infinity,1) U (1,+infinity), (-infinity,+ininity) and [0, +infinity) and the intersection of these sets are [0,1)...

Suppose f(x) = \frac{x^2 - 1}{x-1} . Why do we say that f(x) = \frac{x^2 - 1}{x-1} = x + 1 , if \frac{x^2 - 1}{x-1} isn't defined at x = 1, but (x + 1) is defined at x = 1. I always thought that we say two functions are equal to each other if their equations are the same and their domain...

If P is the projection map from a Riemann domain M \rightarrow C^n, and U is a connected subset of M with P(U)=B, where B is a ball in C^n, then is P injective on U, so it's a homeomorphism on U? P is locally a homeomorphism by definition. It would be related to B being simply connected...

So I'm looking for an example of an infinite integral domain with finite characterestic. That is a infinite integral domain such that there is a prime p such that p copies of any element added together is the additive identity. I'm just looking for a simple counterexample. I'm working...

The Schwartz space on \mathbb{R}^d is defined to be S(\mathbb{R}^d) := \{f\in C^{\infty}(\mathbb{R}^d,\mathbb{C})\;|\; \|f\|_{S,N}<\infty\;\forall N\in\{0,1,2,3,\ldots\}\} where \|f\|_{S,N} := \underset{|\alpha|,|\beta|\leq...