Splitting Definition and 259 Threads

Hi, All: First of all, the title should be "Heegard Splitting of ## S^3 ## ; the 2-torus is not even a 3-manifold. I think I have a way of showing that ## S^3## can be decomposed as the union of two solid tori ## = S^1 \times D^2 ## ,but the argument seems more analytical than geometric...

I am reading Nicholson: Introduction to Abstract Algebra, Section 6.3 Splitting Fields. Example 1 reads as follows: (see attachment) -------------------------------------------------------------------------------------------------- Example 1. Find an extension E \supseteq \mathbb{Z}_2 in...

How do I show Strang splitting has second order accuracy? For the the following equation \[ u_t = (L + M)u, \] where \(L\) and \(M\) are...

I am reading Dummit and Foote Section 13.4 Splitting Fields and Algebraic Closures In particular, I am trying to understand D&F's example on page 541 - namely "Splitting Field of x^p - 2, p a prime - see attached. I follow the example down to the following statement: " ... ... ... so...

Dummit and Foote Exercise 1 on page 545 reads as follows: Determine the splitting field and its degree over \mathbb{Q} for x^4 - 2 . ------------------------------------------------------------------------------------------------------ I have started on the solution to this exercise as...

Can anyone give me some assistance on this problem: I have a pump below a sump of 1m high and pumping upward in a pipe that splits near the end. I require at least 140kPa pressure on EACH pipe. I know I need to enter the pressure in terms of head into the total head (physical height -...

Let $p(x)=x^m-1$ be a polynomial over $\mathbb Q$ and $E$ be the splitting field for $p$ over $\mathbb Q$. We know that $p$ has $\phi(m)$ primitive roots in $E$, where $\phi$ is the Euler's totient function. Let $\omega$ be a primitive root of $p$. Define $\theta_k:E\to E$ as...

I suppose this is less of a "help me with a problem" question than a question asking why something happens. All semester I have been working with vectors and vector components in my general physics class. I understand how to do it and how to solve a complex problem using this method. What I...

Let $f(x)\in \mathbb Q[x]$ be an irreducible polynomial of degree $n\geq 3$. Let $L$ be the splitting field of $f$, and let $\alpha\in L$ be a zero of $f$. Given that $[L:\mathbb Q]=n!$, prove that $\mathbb Q(\alpha^4)=\mathbb Q(\alpha)$. ____ Attempt: Lemma: Let $F$ be any field and $f,g \in...

I'm not really sure when each of these should be done. In fact, I don't really understand the reason that we use the limit comparison test. Σ1/(n^2+1) So here I can simply say that P=2>1, so the original converges. Σ1/N^3+N^2 Here, I would say that P=3>1, implying the original...

I'm reading about path integrals in Peskin and Schroeder's Introduction to Quantum field theory and there is a few things in the text which I find puzzling. At page 283 in the section about correlation functions we are considering the object (equation 9.15) \int D\phi(x) \phi(x_1) \phi(x_2)...

Hello , about the band theory I was confused some how seeing the attached image I ask :- 1- when the atoms are far away we take a single atom alone as a system , but when atoms get closer we identify them all as the system , right ?! 2- why do we use the word " splitting" , why...

Simple and stupid question! A Be atom is traveling with 60 kev kinetic energy and splits into two helium atoms, and the process itself releases 92.2 kev. One helium atom moves at a 30 degree angle with respect to x. find the direction of motion of the second helium atom and find the velocity...

I have read from DJ Griffiths Quantum Mechanics that the electron-positron annihilation process will enlarge the hyperfine splitting of the positronium ground state. More precisely, the actual splitting is larger than that we calculated only considering the spin interaction between the positron...

I am trying to solve a problem that includes a function of the light hitting a certain area. My question is, how would I change a function G(x) of photons hitting a certain area to include just photons of a certain wavelength, say red light. I feel like this could be accomplished using a Fourier...

1. Problem: Consider Hydrogen in its ground state in a magnetic field of magnitude B. Compute the effect of the magnetic field on the hyperfine structure of Hydrogen (that is you should include the interaction of both the proton and electron magnetic moments with the applied magnetic field)...

Hi, Does anyone have an intuitive idea of why it is always the valence bands split under spin/orbit coupling, but not conduction band? (or a much smaller splitting than valence band) I know through tight-binding calculations, if I plug in numbers correctly, conduction bands always have...

Homework Statement Prove the following statement, where A and B are nxn matrices. Tr(AB) = Tr(BA) Homework Equations The Attempt at a Solution Using some manipulations, I arrived at \sum^{n}_{p=1}\sum^{n}_{k=1}A_{pk}B_{kp} = \sum^{n}_{p=1}\sum^{n}_{k=1}B_{pk}A_{kp} If I can prove the...

the polynomial x^4+8x+12=0 has the Galois group A4. I have all its roots, but can't figure out its splitting field. The roots are \alpha_1=\sqrt{2}(\sqrt{\cos{(\pi/9)}}+i\sqrt{\cos{(2\pi/9)}}+i\sqrt{\cos{(4\pi/9)}}) \alpha_2=\sqrt{2}(\sqrt{\cos{(\pi/9)}} -...

http://technet.microsoft.com/en-us/sysinternals/cc817881 That Desktop application by Mark Russinovich helps create new session for desktop (4 in total) but it doesn't seem to be a full *split*. That is, processes created in one session are still alive or known in another session. Do you know a...

A F E D B C Triangle ABC: BC=195, AC=280, AB=323, area=27132 BD=73, CE=80, AF= 34 area triangle BCD = 6132 area triangle CDE = 6000 area triangle DEF = 12960 area triangle AEF = 2040 Big deal (Fubar) BUT: the...

Physicists Detect Elusive Orbiton By "Splitting" Electron "Condensed-matter physicists have managed to detect the third constituent of an electron — its 'orbiton'. Isolated electrons cannot be split into smaller components, earning them the designation of a fundamental particle. But in the...

Homework Statement The question says: Find the degrees of the splitting extensions of the following polynomials, and show that in each case the number of automorphisms of the splitting field is at most the degree of the extension. i) x^3 - 1 over Q (3 others) Homework Equations...

Hi eveyone, I am new to this forum and I hope that I can gain lots from this forum. Recently, one question comes into my mind, that is about the possibility of splitting the electric component and magnetic component of EM wave. This two components seem to always stick together. However, if...

I know that you can split, say H2O by using electrolysis. From there the hydrogen atoms would go to the negative electrode (cathode) and the oxygen atoms would go to the positive electrode (anode). But what if you wanted to separate the two even more, such as pulling the hydrogen atoms to one...

Hi, I am was reading about a double well potential. I came across the word "quantum tunnel splitting". Can anyone tell me what this is? For example if we have a double well potential we can have a |L> and |R> as two states. The article said that if we ignore the tunneling the two states...

Is it possible to split a single light beam into two beams of opposite circular polarization? A properly oriented calcite crystal will separate a unpolarized beam into two beams, one vertically polarized and one horizontally polarized. Other polarizers pass just one polarization and absorb...

Hi, Please forgive any bad forum etiquette - I'm a newbie! I'm trying to figure out a staking system for when I group different bets together into multis, but I can't get one that makes mathematical sense to me. I'll illustrate an example. There are 4 bets to consider: A - Odds: 1.85, True...

We know this tiny splitting in degenerate levels in Hydrogen spectrum due to magnetic interactions between spins of electron and proton. but is that true the proton spin is independent of energy in this (hyperfine) splitting? I want to understand the physics of this splitting and how the spin...

(Herstein Pg 222) DEFINITION: If $f(x) \in F[x]$, a finite extension $E$ of $F$ is said to be a splitting field over $F$ for $f(x)$ if over $E$(that is, in $E[x]$), but not over any proper sub-field of $E$, $f(x)$ can be factored as a product of linear factors. Now here's my question. Take...

I'm working on a project that involves taking the audio output from a consumer device like an iPod or whatever, filtering for low, mid, and high frequencies, and then converting the levels into a PWM to drive LEDs for each frequency range. Passive filters will separate the low (<500 Hz), mid...

Homework Statement From \frac{de_{s}}{dT} = \frac{L_{v}e_{s}}{R_{v}T^{2}} derive e_{s}(T) = 6.11 e^{\frac{L}{RV}(\frac{1}{T}-\frac{1}{273})} Homework Equations The Attempt at a Solution The way my lecturer derived it was he 'split' the derivative and took them to their respective sides...

I wonder if someone could help with me with this. I understand Snells Law and I can also work out various refractions in different media by using C. But “why” does refraction occur. Every website I look at (inc Wiki) gives the results of refraction (e.g. Snell’) not the reason. They will...

So say for example: Somehow you were dislodged from time so whenever you made a decision you never knew what reality you would be in based on your decision. For example, you made a "yes" decision, but would end up in the "no" (opposite of decision) reality, or sometimes you stayed in the...

If $F$ is the field of rational numbers, find the necessary and sufficient conditions on $a$ and $b$ so that the splitting field of $p(x)=x^3+ax+b=0$ has degree exactly $3$ over $F$. ATTEMPT: If $p(x)$ is not irreducible in $F[x]$ then the splitting field of $p(x)$ over $F$ can have degree...

This article says that electron can be split into two: http://www.sciencedaily.com/releases/2012/04/120418134847.htm That means electron is not elementary particle anymore?

I need a quick reminder that this is (hopefully) true: Let \sum a_n be an infinite series of complex terms which converges but not absolutely. Then can we still break it up into its real and imaginary parts? \sum a_n = \sum x_n + i\sum y_n

Hi friends. I'm doing a homework problem about how degenerate orbitals split in the presence of a magnetic field. I understand everything, but I was just had a question about notation. ΔE, the energy between degenerate orbitals in a magnetic field equals μB = (e*hbar/2m)B I was just...

Hi, I've been looking at Fourier transforms, odd and even functions and such recently. But I'm a little confused about how exactly you split a function up. I know the general forumla and seen the derivation, however when i do it with a proper function i never seem to get the correct answer. Was...

In Dummit and Foote, a short exact sequence of R-modules 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 (\psi:A \rightarrow B and \phi:B \rightarrow C) is said to split if there is an R-module complement to \psi(A) in B. The authors are not really clear on what the phrase "an R-module...

I am required to write a program that uses Simpson's rule to evaluate ∫t**-2/3(1-t)**-1/3 dt from limits t=0 to t=1. The questions gives a hint to split the integral into two parts and use a change of variable to handle the singularities. I really don't know where to begin. Is the choice of...

Homework Statement Calculate the energy splitting of the magnetic sub levels of a 3S1 term in a magnetic field of 1.5 T. The Attempt at a Solution From the term symbol 3S1 I can conclude: S -> L=0 subscript 1 -> J=1 superscript 3 -> 3=2S+1 -> S=1 Therefore I have two electrons with spin +1/2...

Does this splitting of d-orbitals happen only in case of coordination compounds or does this happen when transition metals form compounds too? I have only studied about splitting of d-orbitals in case of ligands(i.e. when they form complexes). Does this happen when they form compounds too)...

I want to split the fraction: 2a/((a+1)(a2+4)) I have tried using partial fractions, but came to something that was nonsense, and my question is why that is. Why doesn't partial fractions work in this case, from a mathematical point of view, and is it still possible to split up the fraction...

I need to find the splitting field in \mathbb {C} of x^3+3x^2+3x-4 (over \mathbb{Q} ). Now, I plugged this into a CAS and found that it is (probably) not solvable by radicals. I know that if I can find a map from this irreducible polynomial to another irreducible polynomial of the same...

Exchange splitting of energy levels of a system?? Hey guys, I was just reading my QM book and came across this; Homework Statement Determine the exchange splitting of energy levels of a system of two electrons, regarding the interaction of the two electrons are a perturbation. My...

Homework Statement Consider f(x) = x^3-5 and its splitting field K = Q(5^{1/3}, \omega) where \omega = e^{2 \pi i/3} Show that B = \{1, 5^{1/3}, 5^{2/3}, \omega, \omega 5^{1/3} , \omega 5^{2/3} \} is a vector space basis for K over Q.The Attempt at a Solution I am just a bit confused...

Homework Statement The points a, b and c and mid way along a series of tubes arranged such: There is a tube of 4mm for 100 cm It splits into 3 tubes of diameter 1mm, length 20 cm It converges into a tube of diameter 6mm, length 100 cm The fluid flowing through the tubes is 3ml/sec...

Could someone please explain to me what mass splitting is or ideally provide me a link so I could read about it. Also, how is it related to dark matter or Universal Extra Dimensions Theory? Thanks for any help (I tried google and got nothing).

Hi guys, I'm having a bit of trouble splitting the RHS of the following expression into real and imaginary parts: (χ'+iχ")/A = \frac{1}{ω-ω_{0}-iγ/2} (It's to find expressions for absorption coefficient and index of refraction, but that's irrelevant). I've defined a = ω-ω_{0} and b =...