Extension Definition and 279 Threads

We used to buy these cheap outlet strips that came with really long wires, wound up. They said to unwind them before use because they caused fires. I thought it was inductance. But now that I think about it, there's not just a single conductor with AC in there. There's also ground and...

Can I find out the natural extension of a spring if I am only given the mass of a block that can be put on it and the value of the spring constant? I have found x ( from the formula F = -kx ) when the block is on it but I now need to find the extension of the spring with no mass on the end. It...

Solve U_xx=U_tt with c=1. Dirchlet boundary conditions U(x,0)=1 for 5<x<7 U(x,0)=0 for everywhere else U_t(x,0)=0 I know that by taken an odd extension I can get rid of the boundary condition and then solve the initial value problem using the d'alembert solution and only care for x>0...

Let $T$ be a bounded normal operator and let $x$ be a member of the spectrum. Consider the homomorphism defined on the set of polynomials in $T$ and $T^{*}$ given by $h(p(T,T^*))=p(x,x^*)$ Prove that this map can be continuosly extended to the closure of $P(T,T^*)$

I am attempting to understand Dummit and Foote exposition on 'extending the scalars' in Section 10.4 Tensor Products of scalars - see attachment - particularly page 360) [I apologise in advance to MHB members if my analysis and questions are not clear - I am struggling with tensor products! -...

At the bottom of page 708 in Dummit and Foote (Chapter 15, Section 15.4 Localization) we find the definition of the extension and contraction of ideals. The notation is similar to $$ I^e $$ and $$ I^c $$ except that the superscripts e and c occur before the I. Can someone please help me with...

Broad title, but really a specific question that I thought should be straightforward, but got stuck. Consider the geodesics of form t=contant, r>R, in exterior SC geometry in SC coordinates. These are spacelike geodesics. If we consider this geometry embedded in Kruskal geometry, it is easy to...

I was thinking, if exist a product (cross) between vectors defined as: \vec{a}\times\vec{b}=a\;b\;sin(\theta)\;\hat{c} and a product (dot) such that: \vec{a}\cdot\vec{b}=a\;b\;cos(\theta) Why not define more 2 products that result: \\a\;b\;sin(\theta) \\a\;b\;cos(\theta)\;\hat{d} So, for...

This was an exercise out of Garling's A Course in Galois Theory. Suppose ##L:K## is a field extension. If ##[L:K]## is prime, then ##L:K## is simple. I've developed a habit of checking my work for these exercises religiously (the subject matter is gorgeously elegant, so I want to do it...

[b]1. The problem statement So I am currently working on an velocity/acceleration lab. My physics teacher requires each lab group to find an extension that goes above and beyond the question that we are supposed to answer with the lab. Each group also needs evidence to prove the extension...

1. Homework Statement . Let ##(X,d)## be a metric space, ##D \subset X## a dense subset, and ##f: D→ℝ## a uniformly continuous function. Prove that f has a unique extension to all ##X##. 3. The Attempt at a Solution . I have some ideas but not the complete proof. If ##x \in D##, then...

Problem An ideal spring with spring constant 'k' is hung from the ceiling and a block of mass 'M' is attached to its lower end. The mass is released with the spring initially unstretched. Then the maximum extension in the spring is? Please tell how to do it and the final answer.

I think everybody here knows the equation that gives the potential of a point like dipole, but how does the field look like if you have e.g. a metal sphere with radius $R$ and a certain dipol moment, how does this potential look like?

What is liouville's extension of dirichlet's theorem ? and where can I use such a theorem ? Can I apply Integration like this ? $$\int_0^{\frac{\pi}{2}} \cos^2(x)\sin^2(x)\,dx$$

Let $K$ be a field and $F_1$ and $F_2$ be subfields of $K$. Assume that $F_1$ and $F_2$ are isomorphic as fields. Further assume that $[K:F_1]$ is finite and is equal to $n$. Is it necessary that $[K:F_2]$ is finite and is equal to $n$?? ___ I have not found this question in a book so I don't...

finding the max. extension in the spring...! Okkay so i have to calculate the max. extension in a spring attached with two blocks of mass m and M. The box of mass M is pulled with a force F. The system (blocks of masses m1 and m2 and the mass less spring) is placed on a smooth surface...

Hello, I read that many people believe mathematics to be simply an extension of logic and therefore some or all of math to be reducible to logic. I thought this was an obvious fact for the longest time. I was wondering if there was any flaw with such an argument or what else there is which...

Homework Statement This is from A.P. French, Vibrations and Waves, Problem 3-7 A wire of unstretched length l0 is extended by a distance of 10-3l0 when a certain mass is hung from its bottom end. If this same wire is [turned to be horizontal] and the same mass is hung from the midpoint of...

Firstly, does a stress vs. strain graph for a material always take the same general shape as its load vs. extension graph (with the same important points, e.g. UTS, having the same shape and corresponding to the same thing)? Secondly, what do the stress-strain and load-extension graphs look...

I usually have problems regarding Hooke's law and stuff. Please help me with the question below.. I came across it when I was doing my revision. An explanation will be appreciated. Thanks!

I want to come up with an example of a field extension that is not normal, and seems to be difficult. All extension constructed in some obvious way tend to turn out normal.

[FONT=trebuchet ms]Second question I am stuck on: [FONT=verdana]A spring of natural length l with modulus of elasticity λ has one end fixed to the ceiling. A particle of mass m is attached to the other end of the spring and is left to hang in its equilibrium position under the influence of...

Here is a link to the question: Let E be an extension field of F and let a, b be elements of E. Prove that F(a,b)=F(a)(b)=F(b)(a)? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

I need to determine the correct diameter and length of a combination of 4 extension springs to use in a projectile device. The springs must be able to extend to approx 3ft, and generate enough force to propel a set of objects that are 3ft in length and have an overall weight of approx 5lbs. The...

Let $K=\mathbb{Q}[\omega]$ where $\omega^2+\omega+1=0$ and let $R$ be the polynomial ring $K[x]$. Let $L$ be the field $K(x)[y]$ where $y$ satisfies $y^3=1+x^2$.Which is the integral closure of $R$ in $L$, why?

I was looking at some integration problems the other day and I came across this identity: \int_{0}^{\frac{\pi}{2}} \sin^{p}x \cos^{q}x dx = \frac{1}{2} \mbox{B} \left( \frac{p+1}{2},\frac{q+1}{2}\right) where B(x,y) is the Beta function, for Re(x) and Re(y) > 0. From the way in which the above...

I am using a MAC and have a download with a .msi file extension. Since I do not use Microsoft for my operating system, I can't open the file. Is there a download for this that I can get to open these files?

Prove that the field Q(√2,√3,u) where u^2=(9-5√3)(2√2) is normal over Q. I'm supposed to show that this field is the splitting field of some polynomial over Q. u is clearly algebraic over Q. Do i just take the higher powers of u and try to find the minimal polynomial over Q or is there a...

Hi everyone I 'm having difficulty in proving the following theorem theorem: If L/K ( L is a field extension of K) is a finite extension then it is algebraic. Show, by an example, that the converse of this theorem is not true, in general. Can you help me to find an example in this case? Thanks...

Homework Statement A perfectly elastic spring is attached to the ceiling and a mass m is hanging from the spring. he mass is in equilibrium when the spring is stretched a distance x(o). The mass is carefully lifted and held at rest in the position where the string is nether stretched nor...

Homework Statement The spring block system lies on a smooth horizontal surface.The free end of the spring is being pulled towards right with constant speed v_{0}=2m/s.At t=0 sec the spring of constant k=100 N/cm is unstretched and the block has a speed 1m/s to left.Find the maximum...

Hello, I found this question, and I was able to do the easier parts, but I'm really not comfortable with automorphisms in fields. Let f(x)=x^2 + 1 = x^2 - 2 \in Z_3[x]. Let u= \sqrt{2} be a root of f in some extension field of Z_3. Let F=Z_3(\sqrt{2}). d)List the automorphisms of F which leave...

Experiments show that cosmic ray muons reach Earth surface in greater numbers than they should, unless relativistic time dilation is taken into consideration. It also seems to confirm the SR formula mathematically. However, looking at a lot of different experiment records, I have some doubts...

Hi, All: Just curious as to whether there is some sort of canonical extension of the standard binary connectives: and, or, if, iff, etc. , to n-valued logic. I imagine this may have to see with Lattices, maybe Heyting Algebras, and Order theory in general. Just wondering if someone...

I'm in need of an extension spring with some specific requirments: Material: Stainless Steel Diameter 0.75inch Rest Length: 5 1/2inch inside the hooks At 7 inches a forcle of 10lbs At 10 inches (full extension) a force of no more than 20lbs (As low as possible) Any idea on how to size...

Homework Statement ∫8x3e-cos(x4+4)sin(x4+4)dx Homework Equations Let u = cos(x4+4) The Attempt at a Solution I know the answer does not have the sin in it and only the e remains, because when the integral is found e stays unchanged. I could find somewhere online to calculate it...

Thank you in advance, I need help proving or disproving this. In the binomial theorem, with a power (a+b)^n, I need to prove that a^n + b^n is greater than the rest, or in other words, (a+b)^n - (a^n + b^n).

So if we have an extension of E of F, then we can consider E as a vector space over F. The dimension of this space is the degree of the field extension, I think most people use [E:F]. This is correct in most people's books, right? Defining \omega = cos (2\pi /7) + i sin (2\pi / 7) Why...

I am reading Naive set theory by P R Halmos. He says that "The axiom of extension is not just a logically necessary property of equality but a non-trivial statement about belonging." The example for that is "Suppose we consider human beings instead of sets, and change our definition of...

Is it possible for an extension spring to be 20 inches long while unloaded, 32 or so inches long fully extended(12" max displacement), and have an initial load of 30-40 lbs? All other properties are virtually not important. If so, where could I buy them?

Let F be a field of characteristic 0. Let f,g be irreducible polynomials over F. Let u be root of f, v be root of g; u,v are elements of field extension K/F. Let F(u)=F(v). Prove (with using basic polynomial theory only, without using linear algebra and vector spaces): 1) deg f = deg g (deg f...

Given a differential field F and a linear algebraic group G over the constant field C of F, find a Picard-Vessiot extension of E of F with G(E/F)=G: This isn't homework, just something I saw in a book that I was curious about. The author says that this can be shown but doesn't illustrate how...

i need what we mean by B-L extension of the Standard Model ?

Suppose that E is a field extension of F, and every polynomial f(x) in F[x] has a root in E. Then E is algebraically closed, i.e. every polynomial f(x) in E[x] has a root in E. I've been told that this result is really difficult to prove, but it seems really intuitive so I find that...

Homework Statement A string of length a is stretched to a height of y when it is attached to the origin so making a triangle with length L=\sqrt{a^{2}+\frac{y^{2}}{a^{2}}} and therefore a length extension ΔL= \sqrt{a^{2}+\frac{y^{2}}{a^{2}}}-a which simplifies to...

An experiment consists of tossing a pair of dice: 1) Determine the number of sample points in the sample space 2) Find the probability that the sum of the numbers appearing on the dice is equal to 7 Issue: Ok so I know how to do this problem, but my question comes with respect to the...

Let K be a finite group and H be a finite simple group. (A simple group is a group with no normal subgroups other than {1} and itself, sort of like a prime number.) Then the group extension problem asks us to find all the extensions of K by H: that is, to find every finite group G such that...

Hello, I have a quick question about extension fields. We know that if E is an extension field of F and if we have got an irreducible polynomial p(x) in F[x] with a root u in E, then we can construct F(u) which is the smallest subfield of E containing F and u. This by defining a homomorphism...

Greetings, comrades! In a previous thread, a user articulated a common argument: His analogy mapping knights to horses makes intuitive sense, but how can we apply this idea to two infinite sets of knights and horses? How can we treat finite and transfinite sets equal in that sense and...

Homework Statement The following results were obtained when a spring was stretched: Load/N: 1.0 3.0 4.5 6.0 7.5 Length of spring/cm:12.0 15.5 19.0 22.0 25.0 A) use the results to plot a graph of length of spring against load. b) use the graph to find the: i)...