Extension Definition and 294 Threads

How can the function f: ℝ² → ℝ : (x,y) |--> {{x^2+y^2-(x^3y^3)}\over{x^2+y^2}} if (x,y) ≠ (0,0) be defined in the origin so that we get a continuous function?When I take 'x=y' (so (y,y)) and 'y=x' (so (x,x)) I get: {{2-y^4}\over{2}} and {{2-x^4}\over{2}} So for the first one I get '1'...

We have a trebuchet which we can adjust the launch angle, and how far the rubber band is pulled back (extension). The rubber band spring constant is constant throughout. I need to find the relationship between the angle of launch and range, and the relationship between the extension and range...

Given metric spaces (X, d_X), (Y, d_Y), and subsets \{ x_1, ..., x_n \}, \{ y_1, ..., y_n \} of X and Y respectively, if I define a function that send x_i to y_i, what are the minimal restrictions we need on X and Y in order for there to exist an extension of that map to a continuous...

Based on the results of every variation of the two-slit experiment so far, the presence or absence of the interference pattern is based on whether or not which-path information can be known or not. The Delayed Choice Quantum Eraser experiment...

Homework Statement "Given a function H, assigning a value, 0 or 1, to each atomic sentence, define a sequence ℑ0, ℑ1, ℑ2, ℑ3,... of functions, as follows: ℑ0 is just H. Given a function ℑn, assigning a value, either 0 or 1, to the sentences of degree less than or equal to n, define the...

Homework Statement Find the splitting field of x8 − 2 over Q, including its degreeHomework Equations Degree is multiplicative in tower of fields: F \subset K \subset L [L:F] = [L:K][K:F] Degree of Galois extension is equal to the order of the Galois group [K:F] = |Gal(K/F)| The Attempt at a...

Homework Statement A cylindrical steel bar, 20-mm in diameter and 5.0-m in length, is subjected to an axial tensile load of 40-kN applied at the ends of the steel bar. Determine the axial extension, in millimeters, of the steel bar under this loading condition?Homework Equations E=\sigma/e...

Hi... How do I construct a Galois extension E of Q(set of rational numbers) such that Gal[E,Q] is isomorphic to Z/3Z. Thanks.

Homework Statement A wire of diameter 5.0 mm supports a 2.8kg load. (a) Determine the tension in the wire (b) The original length of the wire was 2.0m Calculate its extension when supporting the load. Homework Equations Young's modulus for the material of the wire = 2.0 x 10^7N...

I am having no luck understanding how to find the basis of a field adjoined with an element. For example Q(sqrt(2)+sqrt(3)) I know that if i take a=sqrt(2)+sqrt(3) that i can find a polynomial (1/4)x^4 - (5/2)x^2 + 1/4 that when evaluated at a is equal to zero. So, from that I know the...

For wave equation: \frac{\partial^2 u}{\partial t^2} \;=\; c^2\frac{\partial^2 u}{\partial x^2} \;\;,\;\; u(x,0)\; =\; f(x) \;\;,\;\; \frac{\partial u}{\partial t}(x,0) \;=\; g(x) D'Alembert Mothod: u(x,t)\; = \;\frac{1}{2} f(x\;-\;ct)\; +\; \frac{1}{2} f(x\;+\;ct)\; +\; \frac{1}{2c}...

First, if there are tests that have ruled one way or the other, that pretty much ends this discussion. I couldn't find any, but if they exist please point them out here. That leaves the discussion of: 1) How could these been distinguished experimentally, and 2) How precise do tests have...

This stuff is killing me... Let K \leq M \leq L be fields such that L is galois over M and M is galois over K. We can extend \phi \in G(M/K) to an automorphism of L to show L is galois over K. I need help filling in the details in why exactly L is galois over K.

Homework Statement Conventionally, the Galilean Transformation relates two reference frames that begin at the same location and time with one reference frame moving at a constant velocity {\vec{v}} along a positive {x}-axis (which is common to both reference frames) with respect to the other...

Two problems from my abstract algebra class... 1) Let K be the algebraic closure of a field F and suppose E is a field such that  F F \subseteq E \subseteq K. Then is K the algebraic closure of E? 2) Let n be a natural number with n\geq2, and suppose that \omega is a complex nth...

Homework Statement The wires on a violin have a cross section of 5.1*10^-7m^2. The wires are put under tension by turning the wooden pegs. The Young Modulus of Steel is 2.0*10^11 Pa. Calculate the tension required to produce an extension of 4*10^-4m. Homework Equations Thats where...

Hi, how can I show that a field extension is normal? Here is a concrete example: L|K is normal, whereas L=\mathbb F_{p^2}(X,Y) and K= \mathbb F_p(X^p,Y^p) . p is a prime number of course. I have to show that every irreducible polynomial in K[X,Y] that has a root in L...

As a consequence of Bezout's identity, if a and b are coprime there exist integers x and y such that: ax + by = 1 The extension states that, if a and b are coprime the least natural number k for which all natural numbers greater than k can be expressed in the form: ax + by Is a+b-1...

field extension question (abstract algebra) Homework Statement For any positive integers a, b, show that Q(sqrt a + sqrt b) = Q(sqrt a, sqrt b). Homework Equations The Attempt at a Solution i proved that [Q(sqrt a):Q] for all n belonging to Z+ is 2 whenever a is not a perfect...

I'm learning about relativity and, by extension, the classic Michelson-Morley setup. I cannot see why a "fringe effect" was expected and could use some discussion. Here is my reasoning. Firstly we imagine the light signal to propagate from the SM (silvered mirror, in the "center") to M1 and...

I've been asked to find out if some field extensisons are normal. I want to know if I'm thinking about these in the right way. For Q(a):Q I first find the minimal polynomial for a in Q[a]. Then I look at all zeros of that polynomial. If all of the zeros are in Q(a) the extension is...

Homework Statement This question is from the Australian HSC maths extension 2 test. Q8b) Let n be a positive integer greater than 1. The area of the region under the curve y=1/x from x=n-1 to x=n is between the areas of two rectangles. Show that...

Homework Statement This problem is from the Australian HSC mathematics extension 2 exam. Q6b) It states: b) Let P(x)=x^3+qx^2+qx+1, where q is real. One zero of P(x) is -1 i) Show that if \alpha is a zero of P(x) then \frac{1}{\alpha} is a zero of P(x) ii) Suppose that \alpha is a...

Homework Statement Allow D to be the circle lz+1l=1, counterclockwise. For all positive n, compute the contour integral. Homework Equations int (z-1/z+1)^n dz The Attempt at a Solution I know to use the extension of the CIF. Where int f(z)/(z-zo)^n+1 dz = 2(pi)i*...

Consider a sping of length, L m and spring constant K N/m. If a force of F N is applied, we will expect the spring to be extended by x m. So The variables(F,K,x) can be combined to formed an equation, known as the hookes law i.e F=Kx. Does the length of spring have any effect on the extension...

I was wondering if the "Method of image charges" could be extended even partly or approximately to oscillating charges. I am not considering nearly-static problems, but really radiating problems. After all, the Poisson equation and the wave equation are rather close ! Therefore, I thought...

Points, extension, time, waves and many times (help!) I have posted this on another forum. Please I am not a spammer, I just mean to find the best informed opinions I have so as to guide me on the path of truth. I have been patiently been trying to think certain riddles through, but the wise...

Homework Statement A light spring of constant k = 85.0 N/m rests vertically on a table (as shown in part a) of the figure below). A 2.25 g balloon is filled with helium (density = 0.180 kg/m3) to a volume of 5.95 m3 and is then connected to the spring, causing it to stretch as shown in part...

Hi all, I have a rod which is part of a greater structure and I resolved the forces along the rod. This rod is under tension with a force FcosQ on the left end (force direction towards left) and FsinQ on the other end (force direction towards the right). I need to find the extension of the rod...

Homework Statement Let p\in\mathbb{Q}[x] be an irreducible polynomial. Suppose K is an extension field of \mathbb{Q} that contains a root \alpha of p such that p(\alpha^2)=0. Prove that p splits in K[x]. The Attempt at a Solution I was thinking contradiction, but if p does not split in...

block of mass 8.0 kg slides from rest down a frictionless 51degree incline and is stopped by a strong spring with The block slides 7.00 m from the point of release to the point where it comes to rest against the spring. When the block comes to rest, how far has the spring been compressed...

Homework Statement I am to illustrate a particular theorem by considering a functional f on R^2 defined by f(x)=\alpha_1 \xi_1 + \alpha_2 \xi_2, x=(\xi_1,\xi_2), its linear extensions \bar{f} to R^3 and the corresponding norms. I'm having a couple problems with this problem. For one, I...

I'm currently trying to prove that (for a field extension K of the field F) if u\in K and u^2 is algebraic over F then u is algebraic over K. I thought of trying to prove it as contrapositive but that got me nowhere--it seems so simple but I don't know what to use for this. Any help with this...

Homework Statement Find a sequence of extension fields (i.e. tower) Q= F_{0}\subseteq...\subseteqF_{n}. where \sqrt{1+\sqrt{2}+\sqrt{3}+\sqrt{5}} \in F_{n} Prove that all the steps are non-trivial. except the last one. btw Q is the set of rational number. and 0 and n on F were...

Hi: a couple of questions on (Alexandroff) 1-pt. compactification: Thanks to everyone for the help, and for putting up with my ASCII posting until I learn Latex (in the summer, hopefully.) I wonder if anyone still does any pointset topology. I see many people's eyes glace when I...

Homework Statement Let f(x) = sin(x)/x for |x| <= pi with the obvious definition at x = 0 Extend it periodically. Will the Fourier series converge at x=0?Homework Equations Fourier coefficients: ao = 1/\pi \int_{-\pi}^{\pi} (sin(x)/x) an = 1/\pi \int_{-\pi}^{\pi} (sin(x)/x) * cos(nx) bn =...

if K/E is a quadratic extension and field F is contained in K such that FE=K and [K:F] is finite, how do I give a non-example to show [F: F intersects E] might not be 2? Thanks a lot!

The spring constant of a helical spring is 28Nm^-1. A 0.40 kg mass is suspended from the spring and set into simple harmonic motion of amplitude 60mm. Calculate (i) the static extension produced by the 0.40 kg mass, (ii) the maximum potential energy stored in the spring during the first...

hi. ok if I'm using the exchange theorem for extension to a basis. i have the standard basis of 4 dimensional real space is {e1,e2,e3,e4}. and v1=e1+e2 then i can say that the coefficient at e1 is 1 which is non zero therefore i can exchange and get {v1,e2,e3,e4} as a basis. however if v2 =...

K=\mathbb{Q}(\sqrt{2+\sqrt{2}}) is a Galois extension of \mathbb{Q} [I showed this]. Determine \text{Gal}(K/\mathbb{Q}) and describe the lattice of subfields \mathbb{Q} \subset F \subset K. I found that \text{Gal}(K/\mathbb{Q})=\mathbb{Z}_4. I do not know how to draw the lattice of subfields...

Homework Statement An ideal spring has a spring constant k = 30 N/m. The spring is suspended vertically. A 1.1 kg body is attached to the unstretched spring and released. It then performs oscillations. (a) What is the magnitude of the acceleration of the body when the extension of the...

My notes say: If K is an extension of F then A={a in K | a is algebraic} is a subfield of K contained in F. But the elements in A need not be in F, right? Shouldn't it be: If K is an extension of F then A={a in K | a is algebraic} is a subfield of K contained in K. But I don't see the...

In algebra I'm having trouble with this definition: The extensions K of F is a simple extension of F if F = F(a) for some a in K. What I understand so far: F is a set. But the notation F(a) has me lost-- what is it? Is is the values you get when you plug a into any polynomial over th field F...

hello, i am currently going over past exam questions to further my knowledge in as physics when i came across this old question which i don't quite understand, a length of steel and a length of brass are joined together. This combination is suspended from a fixed support and a force of 80n is...

Suppose we have a function f from R to R that is continuous on (a,b]. Define g by g(x) = f(x) if x <> a and g(a) = lim f(x) as x approaches a. Is it true that g is continuous on [a,b]? I would think it is, but I'm having a hard time proving it. I'm trying to use sequences to do this: Suppose...

Homework Statement This comes courtesy of Royden, problem 4.14. a.Show that under the hypothesis of theorem 17 we have \int |fn-f| \rightarrow 0 b.Let <fn> be a sequence of integrable functions such that fn \rightarrow f a.e. with f integrable. Then \int |fn-f| \rightarrow 0 if and only...

Homework Statement Let K be a field, and let K' be an algebraic closure of K. Let sigma be an automorphism of K' over K, and let F be the fix field of sigma. Let L/F be any finite extension of F. Homework Equations Show that L/F is a finite Galois extension whose Galois group...

Homework Statement I'm writing a paper on one of my lab experiments and I'm not sure my physics concept is right? The question asks if the mass of a cart affects the extension of the spring. A spring is attached to a cart which is attached to sting through a pulley and connected to a hanging...

Does anybody know, how to find the derivative of the F with respect to a? As far as I know the Leibnitz rule is only applicable, when the integration limits do not depend a. But what happens, when one of the limits is a function of a? F(a,x)=\int ^{c+h(a)}_{c} f[g(a,x)] dx Thank you so much!

A proposed MDM goes beyond the standard model in a minimal way, so as to produce a candidate for dark matter. Here is story from Nature News: http://www.nature.com/news/2008/080902/full/455007a.html Here is a key excerpt: ==quote== ...It now seems that some physicists have taken...