Chain Definition and 939 Threads

I am reading Joseph J. Rotman's book: Advanced Modern Algebra (AMA) and I am currently focused on Section 7.1 Chain Conditions (for modules) ... I need some help in order to gain a full understanding of some remarks made in AMA on page 526 on modules in the context of chain conditions and...

Homework Statement Homework Equations Chain rule, partial derivation The Attempt at a Solution dv/dt=dv/dx*dx/dt+dv/dy*dy/dt dx/dt=-4t -> evaluate at (1,1) =-4 dv/dt=-4dv/dx+4(-2) dv/dt=-4dv/dx-8 How can I find the missing dv/dx in order to get a value for dv/dt? Thanks!

Say I have a function of three variables, ##F=F(s_{12},s_{23},s_{13}) = F(s,t,-s-t)##, where ##s_{12}=s,s_{23}=t## and ##s_{13}=u = -s-t##. I want to compute the differential operators $$\frac{\partial}{\partial s}, \frac{\partial}{\partial t}\,\,\text{and}\,\,\frac{\partial}{\partial u}.$$...

Homework Statement Homework Equations dz/dt = dz/dx⋅dx/dt + dz/dy⋅dy/dt The Attempt at a Solution [/B] I am getting : =[-sin(x+7y) ⋅ 10t] + [-sin(x+7y) ⋅ 7 ⋅ (-1/ t2)] then changing x and y terms: =[-sin((5t2)+7(1/t)) ⋅ 10t] + [-sin((5t2)+7(1/t)) ⋅ 7 ⋅ (-1/ t2)]

I am reading P.M. Cohn's book: Introduction to Ring Theory (Springer Undergraduate Mathematics Series) ... ... I am currently focused on Section 2.2: Chain Conditions ... which deals with Artinian and Noetherian rings and modules ... ... I need help with understanding a feature of the Theorem...

Homework Statement For the following Markov chain, find the rate of convergence to the stationary distribution: \begin{bmatrix} 0.4 & 0.6 \\ 1 & 0 \end{bmatrix} Homework Equations none The Attempt at a Solution I found the eigenvalues which were \lambda_1=-.6 or \lambda_2=1 . The...

Homework Statement A loop of flexible chain, of total weight W, rests on a smooth, frictionless right circular cone of base radius r and height h. The chain rests in a horizontal circle on the cone, whose axis is vertical. Find the tension in the chain. Homework Equations Virtual work, but...

Im stuck on theorem 5 where the book used chain rule then used product rule then again using the chain rule. How in the world does it work? I don't get product rule used and chain rule used after.

Homework Statement Consider a system made up of joining together ##N## beads and ##N-1## springs. The positions of the beads is indicated by ##N## real numbers ##\left\{x_i\right\}_{i=1,...N}.## The Hamiltonian which characterises it is $$\mathcal H =...

1. The problem statement Given a stochastic matrix P with states s_1...s_5: P = \begin{pmatrix} 1 & p_2 & 0 & 0 & 0\\ 0 & 0 & p_3 & 0 & 0\\ 0 & q_2 & 0 & p_4 & 0\\ 0 & 0 & q_3 & 0 & 0 \\ 0 & 0 & 0 & q_4 & 1 \end{pmatrix} and the matrix A (which is obviously related to P, but I can't see...

Homework Statement Chain of mass M hangs between 2 walls with its ends at the same height. The chain makes an angle θ with each wall. Find the tension at the lowest point of the chain. a) By considering the forces on half of the chain. b) By using the fact that the height of the chain is...

Hello Forum, I have a couple of kinematics questions. The position of a point object is given by the position vector x(t). Speed is v(t)=dx(t)/dt and the acceleration a(t)= dv(t)/dt. What if we wanted to know the velocity and/or the acceleration as a function of position, i.e v(x) or a(x)...

As I know, when a link in a kinematic chain is fixed, the kinematic chain will become a mechanism. And, I only know the definition of the DOF of a mechanism. Is there the definition of the DOF of a kinematic chain ?

Homework Statement 1. (30 points) A very flexible uniform chain of mass M and length L consisting of very small links is suspended from one end so that it hangs vertically, the lower end just touching the surface of a table. The upper end is suddenly released so that the chain falls onto the...

Hey all, I am reading Goldstein and I am at a point where I can't follow along. He has started with D'Alembert's Principle and he is showing that Lagrange's equation can be derived from it. He states the chain rule for partial differentiation: \frac{d\textbf{r}_i}{dt}=\sum_k \frac{\partial...

Homework Statement z = ƒ(x,y), x = rcos(θ), y = rsin(θ) Use the chain rule to show that: \frac{1}{r^{2}}\frac{\partial ^{2} z}{\partial \theta ^{2}} = sin^{2}(\theta)\frac{\partial ^{2} z}{\partial x^{2}}-2sin(\theta)cos(\theta)\frac{\partial ^{2} z}{\partial x \partial...

Homework Statement Figure shows a massless wheel of radius R on which at a point a mass m is fixed and a uniform chain of mass 2m is tied to it which passes over the rim of the wheel and half of its length is hanging on other side as shown in the figure. When a small clockwise jerk is given to...

Homework Statement Markov process has probabilities p_{j,j+1} = 1-p_{j,0} = (\frac{j+1}{j+2})^k for j=0,1,2,... If T_j = min[n>1 : X_n=j] What is E[T(j)|X_0=j] for j=0,1,2,...? Homework EquationsThe Attempt at a Solution I figured that T(j)|X_0=j is j+1 but don't know how to work out the...

Hello there, I'm starting S382 astrophysics with the OU. The course book says "The proton-proton chain converts four hydrogen nuclei (protons) into a ^4_2He nucleus, two positrons that quickly collide with electrons and are annihilated, and two neutrinos. Hence, branch I of the p-p chain may be...

Homework Statement Homework EquationsThe Attempt at a Solution I have the solution to this problem and the issue I'm having is that I don't understand this step: Maybe I'm overlooking something simple but, for the red circled part, it seems to say that ∂/∂x(∂z/∂u) =...

Hi, I'm a beginner in mechanics so I need some help with a project I started. My project consists of a small kart-like car. The car will be Rear-Wheel-Drive and Rear-Engine too (RR layout). and I am willing to use a small 110cc moped engine. I have a trike Differential Gear for the rear shaft...

< Mentor Note -- OP has been reminded to use the HH Template and show their work in future posts... >[/color] A man fires a bullet into a swimming pool. There is a distance of 6m from the gun to the bottom of the pool. Does the bullet have enough force through the water to break an average...

Homework Statement I am trying to solve the model analitically just for 2 sites to have a comparison between computational results. The problem is my professor keeps saying that the result should be a singlet ground state and a triplet of excited states, but when I compute it explicitally I...

Hi I'm learning about The chainrule, and I understand how to apply the chain rule on various problems, but there is a problems I don't understand how works: 1) The book I'm reading writes acceleration as a=v*(dv)/dt And IT argues that v=ds/dt and a=dv/dt (which i understand) So therefore by...

Homework Statement A uniform chain of length L = πR and mass M is placed on the upper half of a uniform thin disc of radius R and mass M. The disc is placed vertically and can rotate freely about its center that is fixed in space. With a small disturbance the chain starts to fall. Find out the...

Hello! (Wave) A family of sets is called chain if any two sets of the family are comparable, i.e. the first set contains the second or the second the first. How many sets can a chain of subsets of an infinite set contain? More specifically, is there an infinite set, each chain of which is...

Homework Statement I'm unable to see fully how the following equality is determined: $$\frac {\partial f(y + \alpha \eta, \, y' + \alpha \eta', \, x)}{\partial \alpha} = \eta \frac {\partial f}{\partial y} + \eta' \frac {\partial f}{\partial y'}$$ where ##y = y(x)##, ##\eta = \eta (x)##, and...

Homework Statement The polyacetylene chain is a 1D chain of Carbon atoms with single bonds and double bonds in succession. Spacing for single bond is ##a_s = 0.144~nm## and spacing for double bond is ##a_d = 0.136~nm##. Describe the structure using a "lattice" and a "basis". Sketch the...

Homework Statement Ok, don't get angry with me. The original problem is from Solid State Physics but my problem is very well in "Introductory physics". Here is the problem: The chain consists of molecules, which has three atoms, each with mass ##M##. Spring constant between the atoms inside...

Hi. Here's the dispersion relation for a diatomic linear chain, where the distance is a/2 between each atom. My issue here is that if you set m_1=m_2=m, i.e. set both atoms equal to each other, it doesn't automatically reduce to the old acoustic dispersion relation as the ± term doesn't...

Hello all, I have two rollers that will be rotated which will in turn rotate another tube on top of them. The connection will be made with a drive chain between the motor sprocket and the roller sprocket(s). My question is, would it be better to connect the chain to one roller, or to both? I'd...

Homework Statement Homework EquationsThe Attempt at a Solution I sort of want to make this into a linear program problem, but I think that it should be solvable without it, since I never learned about it in this particular course. I will just work with Fuel Process 1 as an example, and I can...

Homework Statement [/B] hi could some body please help me factorise this please ? any chance of a few stages would be much appreciated Homework EquationsThe Attempt at a Solution my attempt , but my solutions say otherwise ? [/B]

Homework Statement Consider the function of two variables: u(x,y) = f(x-y) + g(x+(1/3)y) where f(s) and g(t) are each arbitrary functions of a single variable. Using the change of variables: s = x-y t = x-(1/3)y use the chain rule to determine the first and second derivatives of u with...

I'm in Calc 1 and the Chain Rule is giving me one hell of a rough time. I've spent about 10-12 hours over the last few days just on the homework problems in this one section (only getting about 15-20 problems done) and still feel like I barely understand it. Does anyone have any tips, tricks...

Homework Statement ##J=r^{2}\dot{\phi}## [1] ##\dot{r^{2}}=E^{2}-1-\frac{J^{2}}{r^{2}}+\frac{2MJ^{2}}{r^{3}}+\frac{2M}{r}##. [2] (the context is geodesic equation GR, but I'm pretty sure this is irrelevant). where ##u=r^{-1}## Question: From these two equations to derive...

Homework Statement Homework Equations W = F d F = ma The Attempt at a Solution so in order to get the whole chain on the table, we need to pull the chain 0.65 meters onto the table. since 0.65 meters is hanging off the table, the gravity is acting on it, therefore F=ma where m is half the...

Homework Statement Given that the surface x^7y^2+y^4z^6+z^8x^8+9xyz=12 has the equation z=f(xy) in a neighbourhod of the point (1,1,1) with f(x,y) differentiable, find the derivatives. df/dx (1,1) = ? d^2f/dx^2 (1,1) = ? Homework EquationsThe Attempt at a Solution df/dx (1,1) I got -24/23 or...

When can I do the following where ##h_{i}## is a function of ##(x_{1},...,x_{n})##? \frac{\partial}{\partial x_{k}}\frac{\partial f(h_{1},...,h_{n})}{\partial h_{m}}\overset{?}{=}\frac{\partial}{\partial h_{m}}\frac{\partial f(h_{1},...,h_{n})}{\partial x_{m}}\overset{\underbrace{chain\...

Homework Statement Consider the transverse field Ising model, with $$H=-J\sum_i\left(\sigma^x_i\sigma^x_{i+1}+g\sigma^z_i\right)$$ I have to calculate the magnetization $$\langle\sigma_z\rangle$$ at finite temperature. Homework EquationsThe Attempt at a Solution I have to say, I'm a bit lost.

Hi, I need urgent answers. Basically, I don't have background in Markov and I don't need to learn it now actually. But I have to solve the questions below somehow. If somebody can give detailed answers to the questions below (From beginning to the final solution with explanations), then I will...

Homework Statement A certain laid-back runner owns three pairs of shoes, which he keeps by either the front or back doors of his house. Each morning he is equally likely to leave through the front or back door, and then after his run, he is equally likely to enter through the front or back...

Homework Statement y = (tan^-1(6x))^2 Homework Equations Chain Rule, power rule? The Attempt at a Solution Okay, so I did power rule to bring it to 2(tan^-1(6x)) Then, I know to use the chain rule... I get 2(tan^-1(6x)*(1/1+x^2)... I know u = 6x so I play 6 into x^2 and I get 6x^2... I see...

Hi all I don't really understand this... How come that if an amino group is attached to the amino acid side chain, like in arginine or lysine, the molecule is basic, but if an hydroxyl group is attached, like threonine, it is not basic? How come the amino group can accept a H+ and a hydroxyl...

I'm looking for a simple latch design. My project will have two roller chain sprockets positioned one above the other with about a two foot gap. A motor will turn one of the sprockets and the other is just an idler sprocket. A #40 roller chain will run around the two sprockets. Mounted just...

Consider a function of several variables ##T=T(x_{1},...,x_{3N})## Let's say I have N vectors of the form ##\vec{r_{1}}=(x_1,x_{2},x_{3})## and ##x_j=x_j(q_1,...,q_n)##. Awkward inex usage but the point is just that the each variable is contained in exactly 1 vector. Is it correct to in...

Hello, I solved this exercise, but I probably did mistake in simplification... f(x)=${\left(-2{x}^{2}+3\right)}^{4}$${\left(9{x}^{2}+7\right)}^{12}$ They asked to find derivative, so here is what I did...

So i have an equation problem that i need to find the 2nd derivative of, but my understanding of the chain rule is not the best. I tried working it out but i don't know if i did it correctly. i was given the equation y=4(x2+5x)3 So to take the first derivative, i started off by using the chain...

I need help in understanding the proof of the Chain Rule for differentiation, as presented in Theorem 5.1.6 in Manfred Stoll's book: Introduction to Real Analysis. Theorem 5.6.1 in Stoll (page 173) reads as follows:In the above proof we read the following: " ... ... By identity (3) and then...

Homework Statement Calculate the activity of ##^{222}Rn## in an ore sample containing 5g of natural uranium. Homework Equations ##^{238}U## decay chain (to Radon): ##^{238}U\rightarrow^{234}Th\rightarrow^{234}Pa\rightarrow^{234}U\rightarrow^{230}Th\rightarrow^{226}Ra\rightarrow^{222}Ra##...