Exercise — 400 discussions

Exercise 2.1.5 in Berrick and Keating: An Introduction to Rings and Modules reads as follows: Let $$M$$ be an abelian group with $$Mc = 0$$ for some positive integer $$c$$, and put $$c = ab$$ for coprime integers $$a,b$$. Write $$1 = ar + bs$$, and define endomorphisms $$\alpha$$ and $$\beta$$...

I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K). I need help with PART 2 of Exercise 1.1.11 (Chapter 1: Basics, page 33) concerning bimodules and endomorphisms... ... Exercise 1.1.11 reads as follows...

I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K). I need help with Exercise 1.1.11 (Chapter 1: Basics, page 33) concerning bimodules and endomorphisms... ... Exercise 1.1.11 reads as...

I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K). I need help with Exercise 1.2.8 (a) (Chapter 1: Basics, page 3o) concerning $$K^n$$ as a $$K[T]$$-module ... ... First, so that MHB readers will understand the relevant notation for...

I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K). I need help with Exercise 1.1.4 (iii) (Chapter 1: Basics, page 12) concerning matrix rings ... ... Exercise 1.1.4 (iii) (page 12) reads as follows: I can show that $$M_n (\alpha)$$...

I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K). I need help with Exercise 1.1.4 (ii) (Chapter 1: Basics, page 13) concerning matrix rings ... ... Exercise 1.1.4 (ii) (page 13) reads as follows: I need help with Exercise (ii)...

For those who have read Halmos, in Section 6 Ordered Pairs (page 23 in my book), he gives a non-trivial exercise to find an intrinsic characterization of those sets of subsets of A that correspond to some order in A. I'm curious what that characterization is. A is suppose to be a quadruple...

Hey! :o I have the following exercise: If $p$ is prime, $p \nmid a$, $p \nmid b$, prove that $$a^p \equiv b^p \pmod p \Rightarrow a^p \equiv b^p \pmod {p^2}$$ My idea is the following: $$ a^p \equiv b^p \pmod p $$ $$ a^p \equiv a \pmod p $$ $$ b^p \equiv b \pmod p $$ $$ a \equiv b \pmod p...

For the diagram overleaf solve the problems by at least two different methods. Firstly use you knowledge of the equations of motion for motion under gravity where air resistance is neglected. Secondly, knowing the shape of the curve, find the relationship between height and horizontal...

The problem statement Let ##A ∈ K^{m×n}## and ##B ∈ K^{n×r}## Prove that min##\{rg(A),rg(B)\}≥rg(AB)≥rg(A)+rg(B)−n## My attempt at a solution (1) ##AB=(AB_1|...|AB_j|...|AB_r)## (##B_j## is the ##j-th## column of ##B##), I don't know if the following statement is correct: the columns of...

Homework Statement . Let ##A \in \mathbb C^{m\times n}##. Prove that tr##(A^*A)=0## if and only if ##A^*A=0## (here ##0## obviously means the zero matrix). The attempt at a solution. By definition of the trace of a matrix, the implication ← is obvious. I am having problems proving...

Homework Statement Homework Equations That is my solution: The Attempt at a Solution i solve my problem without ø=30°, I confused what this angle was used to do?

Hi everyone , this exercise was given in one of my midterms , but we didn't correct it and I'm wondering where I went wrong on it : Help will be extremely appreciated : Here is the statement : A block of mass m=2 kg is pushed by a spring with a spring constant of k=650 N/m which is...

A uniform solid sphere of mass M and radius R is rolling without sliding along a level plane with a speed v = 2.30 m/s when it encounters a ramp that is at an angle θ = 27.6° above the horizontal. Find the maximum distance that the sphere travels up the ramp if : 1- the ramp is frictionless...

Hi everyone, my physics final is coming in 3 days:cry: , and I really need to have an answer to this exercise , but I'm stuck ! I don't even understand the problem statement HELP ! A typical fastball is thrwon at approximately 90 mph and with a spin rate of 113 rpm (I don't understand what it...

Jorrie's calculator (Lightcone) makes cosmic history tables which tell you among other things the Hubble times in past years. For convenience let's temporarily use greek Theta Θ to stand for THubble so we don't have to write so much. Basic facts (definitions actually) are that Θ = 1/H and the...

Hi everyone , I'm really stuck with an exercise I can't solve any question and it is really important ! I need help : Two identical billiard balls start at the same height and at the same time and roll along different tracks,as shown in figure attached 1- Which ball has the highest speed...

Given the following premises: {¬p→r∧¬s, t→s, u→¬p, ¬w, u∨w} The conclusion is said to be: ¬t∨w Here are my steps. My conclusion is different from the supposed one, therefore I would appreciate it if any of you can point out my error. Thank You. 1 ¬p→(r∧¬s) Premise 2 p∨(r∧¬s) Implication law...

This is a question for those with Shankar in their disposal. When studying from this book, Exercise 5.1.1 puzzled me. Now, it is easy to solve the exercise from 5.1.9. However, how would we expand psi in terms of the energy eigenfunctions,* if we did not have 5.1.9*, but instead started fresh...

I know that we use fat and other molecules to store energy, and that during exercise we use that energy. I understand that we use that energy by means of a chemical reaction that, among other things, transforms fat in something else. What I wonder is: what is fat ultimately transformed to...

Here is the problem statement: I thought it's a straightforward exercise on the divergence theorem, yet it looks like \operatorname{div} f = 0 . So the surface integral is zero? Am I missing some sort of a trick here? The exercise isn't supposed to be that easy. Any hints are very appreciated!

Hey! :o I got stuck at the following exercise... Could you give me an idea how to show this? Let $G=(V,E)$ be a connected graph and $u,v$ $\epsilon$ $V$.If $d(v,u)=k$,then there is a path $v=v_{1},v_{2},...,v_{k+1}=u$ so that $\{v_{i},v_{j}\}$ doesn't belong in $E$ for $j \geq i+2$...

Homework Statement Has anyone solved the part (d) of 5.6 problem of that book? I am unable to solve it. It asks the reader to prove that the radius ##R## of a rotating cylinder (rotating around its symmetry axis) has to be greater or equal than ##\frac{|S|}{ M } ##, in other words...

I'm looking for some advice for my final year project at university. My project is a design of an exercise machine for wheel chair users . The machine will look like a standard wheel chair but the wheels are a couple of centre meters off the floor. I want to add a mechanism to the wheels which...

I am reading R.Y. Sharp's book "Steps in Commutative Algebra" At the moment I am trying to achieve a full understanding of the mechanics and nature of LEMMA 1.11 and am reflecting on Exercise 1.12 which follows it. LEMMA 1.11 reads as follows: (see attachment)...

hi there, In this Ex ( see attached snapshot ), point b), the poisson bracket equation is not so straightforward to obtain. Please correct my Poisson Bracket expansion here : The first one which is provided is simpler : [ε,η] = εμδμηρ - ημδμερ = ζη and the monster one : [pε,pη] =...

Hello! I need some help at the following exercise: The language L={l ε {a,b}*:the word l does not contain the subword bba} is regular.Which are the equivalence classes of the relation \approx_{L} ? Also,which is the smallest(as for the number of states) deterministic automata that recognize...

I am reading Dummit and Foote Section 3.1: Quotient Groups and Homomorphisms. Exercise 17 in Section 3.1 (page 87) reads as follows: ------------------------------------------------------------------------------------------------------------- Let G be the dihedral group od order 16. G = <...

Dummit and Foote Section 15.1, Exercise 24 reads as follows: --------------------------------------------------------------------------------------------------------- Let V = \mathcal{Z} (xy - z) \subseteq \mathbb{A}^3 . Prove that V is isomorphic to \mathbb{A}^2 and provide an explicit...

Dummit and Foote Section 15.1, Exercise 24 reads as follows: --------------------------------------------------------------------------------------------------------- Let V = \mathcal{Z} (xy - z) \subseteq \mathbb{A}^3 . Prove that $$ V $$ is isomorphic to \mathbb{A}^2 and provide an...

Dummit and Foote (D&F), Ch15, Section 15.1, Exercise 15 reads as follows: ---------------------------------------------------------------------------------------------------- If k = \mathbb{F}_2 and V = \{ (0,0), (1,1) \} \subset \mathbb{A}^2 , show that \mathcal{I} (V) is the...

Dummit and Foote (D&F), Ch15, Section 15.1, Exercise 15 reads as follows: ---------------------------------------------------------------------------------------------------- If k = \mathbb{F}_2 and V = \{ (0,0), (1,1) \} \subset \mathbb{A}^2 , show that \mathcal{I} (V) is the...

I am reading Dummit and Foote Chapter 15, Section 15.1: Noetherian Rings and Affine Algebraic Sets. Exercise 10 reads as follows: -------------------------------------------------------------------------------------------------------------------- Prove that the subring: k[x, x^2y, x^3y^2...

I am reading R. Y. Sharp: Steps in Commutative Algebra, Chapter 5 - Commutative Noetherian Rings Exercise 8.5 on page 147 reads as follows: -------------------------------------------------------------------------------------------------- 8,5 Exercise. Show that the subring \mathbb{Z} [...

Homework Statement could you please check if this exercise is correct? thank you very much :) ##f(x,y)=\frac{ |x|^θ y}{x^2+y^4}## if ##x \neq 0## ##f(x,y)=0## if ##x=0## where ##θ > 0## is a constant study continuity and differentiabilty of this function The Attempt at a Solution...

We're usually just told that exercise is a great way to stimulate the movement of glucose into cells. But how does it do that? I would think that insulin resistance is insulin resistance, exercise or no exercise.

Hi, please refer to the attached image. I am having trouble when doing Exercise 2 Here is what I did: $ \int_{-2}^{2}(f(x)\sin(\frac{m\pi x}{2}))dx = \sin(\frac{m\pi x}{2})a_{0} + \int_{-2}^{2} \sum\limits_{n=1}^\infty (a_{n}(\cos(\frac{n\pi x}{2})\sin(\frac{m\pi x}{2})+b_{n}\sin(\frac{n\pi...

By that I mean, are there any weight-lifters on PF? Women/men, squats/curls, anyone pretty into exercise and nutrition?

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...

Let Fn denote the Fibonacci sequence. un is the sequence given by: un= Fn+1/Fn. Show that mod(un - \phi) \leq\frac{1}{\phi}mod(un-1-\phi) and therefore mod(un - \phi) \leq \frac{1}{\phin-1}[/itex]mod(u1-\phi) and then conclude un converges to \phi I have tried with the identity \phi = 1+...

------------------------------------------------------------------------------------------------ 7. Prove that \mathbb{Q} ( \sqrt{2} + \sqrt{3} ) = \mathbb{Q} ( \sqrt{2} , \sqrt{3} ). Conclude that [\mathbb{Q} ( \sqrt{2} + \sqrt{3} ) \ : \ \mathbb{Q} ] = 4 . Find an irreducible...

Can someone help me get started on the following problem. Determine the degree over \mathbb{Q} of \ 2 + \sqrt{3} and of 1 + \sqrt[3]{2} + \sqrt[3]{2} Peter [This has also been posted on MHF]

Dummit and Foote Exercise 2, Section 13.2, page 529 reads as follows: ------------------------------------------------------------------------------------------------------------------ 2. Let g(x) = x^2 + x -1 and let h(x) = x^3 - x + 1 . Obtain fields of 4, 8, 9 and 27 elements by...

Dummit and Foote Chapter 13, Exercise 2, page 519 reads as follows: "Show that x^3 - 2x - 2 is irreducible over \mathbb{Q} and let \theta be a root. Compute (1 + \theta ) ( 1 + \theta + {\theta}^2) and \frac{(1 + \theta )}{ ( 1 + \theta + {\theta}^2)} in \mathbb{Q} (\theta)...

Hello, in exercise 11, I get b is not and w is, but the answer key says b is and w is not...? I don't understand...have I misunderstood or is the answer key wrong? Regrds, $$|\pi\rangle$$

I am studying Dummit and Foote Chapter 13: Field Theory. Exercise 1 on page 519 reads as follows: =============================================================================== "Show that p(x) = x^3 + 9x + 6 is irreducible in \mathbb{Q}[x] . Let \theta be a root of p(x). Find the...

Homework Statement If anyone could help me with this classical mechanics exercise I would be very grateful! The exercise is as follows: The muon (μ) is a particle with mass mμ=207me, with me being the electron mass. The pion (∏) has a mass of m∏=273me. The pion can decay into a muon...

Hello, Homework Statement Suppose we have two boxes, numbered 1 and 2. Box 1 contains 10 white and 6 numbered red balls, while Box 2 contains 8 white and 12 numbered red balls. We take out 2 balls from Box 1 and are transferred in Box 2. Then, we choose 1 ball from Box 2. a) Find the...

Hello, I require some assistance with this problem. It reads: If, u=1/2x^3; v=3(u+u^2); w=1/v^2; Find dw/dx. It's clear that you should use the chain rule so I begin by calculating the derivative for each one. I get v^(-2); 6u+3; and 3/2x^2. I then have to multiply them but this is where...

Hi, I tried to solve this problem myself and I'd like someone to check it :) Thanks already! A ball is rolling towards a block, which is connected to a spring. Assume no friction occurs. The initial velocity of the ball is 10 m/s. The spring constant is k = 5 N/m. Mass of the ball = 5 kg and...