Constraints Definition and 217 Threads

Hello, I am working on a research project that requires me to write a solver for solving a particular problem. I could really use some math advice if anyone is willing to assist. I need to minimize a non-linear objective functions of 5 variables. It is a pretty complex function. Each of the...

So I need to find the min and max values of f(x,y,z) = x^2 + 2y^2 + 3z^2 given the constraints x + y + z = 1 and x - y + 2z =2. I've gotten as far as (2x, 4y, 6z) = (u,u,u) + (m,-m,2m). I'm stuck trying to solve this system of equations. Any hints?

Please correct me if I make any mistakes along the way. Suppose we have a simple tight-binding Hamiltonian H=\sum_i \epsilon _i c_i^\dagger c_i - t\sum_{\langle i j\rangle} c_i^\dagger c_j + h.c., In half-filling systems, we tend to impose a constraint such that each site has only one...

Homework Statement Using the corresponding constraints conditions, calculate the kinetic energy of 1)A homogeneous cylinder of radius a that rolls inside a cylindrical surface of radius R>a.Homework Equations My toughts: I hope they meants "roll without slipping". Let's consider this case...

http://www.math.northwestern.edu/~clark/285/2006-07/handouts/lin-constraint.pdf It's actually proof of finding sign definiteness of quadratic form with linear constraints with sign of submatrices of bordered hessian. The proof is from page 2~page 3. I have 2 questions: 1. From about...

Hi, I am working on a project for my research and am need of some advice. My background is in computer engineering / programming so I'm in need of some help from some math people :) I need to use steepest descent to solve a problem, a function that needs to be minimized. The function has 5...

Homework Statement Find the extreme values of the function f(x,y,z) = xy + z^2 in the set S:= { y\geq x, x^2+y^2+z^2=4 } Homework Equations The Attempt at a Solution Ok, so This is clearly a lagrange multiplier question. Geometrically, I can see that the region that is the constraint is...

Hi, I am working on a calculus of variations problem and have a general question. Specifically, I was wondering about what kind of constraint functions are possible. I have a constraint of the form: f(x)x - \int_{x_0}^x f(z) dz \leq K If I had a constraint that just depends on x or...

Hello, I am considering the set of all (differentiable) even functions with the following properties: 1) f(x)=f(-x) 2) f(0)=a_0, with a_0\in \mathbb{R} 3) f(n)=0, where n\in \mathbb{Z}-\{0\} One example of such a function is the sinc function sin(\pi x) / \pi x. Is it possible to find...

Hey all, A friend asked me for help the other day on his QM homework. The problem regards a rectangular potential U(x) = \begin{cases} V_0 & -a \leq x \leq a \\ 0 & \text{otherwise} \end{cases}, \qquad E<V_0 I thought about this for a while and checked a few textbooks. If we solve this in a...

Problem: A = n by m matrix x = m by 1 vector y = n by 1 vector C = c by m matrix E = e by m matrix Alpha, gamma and theta are constants. norm(Ax-y) = min subject to: norm(Cx) = alpha norm(Ex) = gamma transpose(Cx)*Ex = (alpha^2)*(gamma^2)*cos(theta) I read a paper on how to do this with 1...

Homework Statement Consider the intersection of two surfaces: an elliptic paraboloid z = x2 + 2x + 4y2 and a right circular cylinder x2 + y2 = 1. Use Lagrange multipliers to find the highest and lowest points on the curve of intersection The Attempt at a Solution L = x^2 + 2x + 4y^2...

Homework Statement Given a vector z=<-12, 1, 1, 2, 7, 0> in R^6 and z=u+v, then find u and v such that u's coordinates are all equal to each other (like <0,0,0,0,0,0>) and v has coordinates that add up to 0Homework Equations z=u+vThe Attempt at a Solution i have no idea how to approach...

Hello everybody. I have a question about the physical meaning of the simplicity constraints that is often used in spin foams. For example, in http://arxiv.org/PS_cache/arxiv/pdf/1004/1004.1780v4.pdf, eq. (34), it is written as K=-\gamma L where K are the boost and L the rotations. Is...

In hamiltonian formulation of GR there appears some constraints (it may be found e.g. in "Modern canonical quantum GR" by Theimann, ch. 1.2). I would like to find a Dirac algebra of the constraints (i.e. compute Poisson bracket between constraints), but my results are not consistent with...

How do pulleys help in carrying load? may be to reduce efforts.

Just read an article on Universe Today at: http://www.universetoday.com/77662/missing-milky-way-dark-matter/ The paper is at http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.1289v1.pdf . It introduces what seems to be rather tight constraints on the dark disc. Any cosmologists care to...

Homework Statement Which of these bodies has redundant constraints for the given loading conditions? F_1 and F_2 are applied, known forces. In the first choice, the support at A is fixed and a cable connects points B and C. In the second choice, the support at A is a smooth pin, and a cable...

Homework Statement A person who is properly constrained by an over-the-shoulder seat belt has a good chance of surviving a car collision if the deceleration does not exceed about 30 "g's" (1.0 g = 9.80 m/seconds squared). Assuming uniform deceleration of this value, calculate the distance...

Hello I am using Landau's mechanics Vol I for classical mechanics. On page 4 he mentions for Lagrangian of a system composed of two systems A and B which are so far away so that their interactions can be neglected. then for the combined system we have L = LA + LB I'm trying to...

Hi, I have a short question about gauge theories and constraints. Imagine I have a symmetry algebra, and I gauge it. With N generators in the algebra I get N gauge fields and N gauge curvatures. In realizing the algebra on the gauge fields I assume the gauge parameters are independent and...

Hi, I have a question about imposing constraints in order to obtain theories of gravity from gauge algebras. Let's take as a warming-up Poincare gravity. The procedure is as follows: * Gauge the Poincare group with generators P (translations) and M (rotations) to obtain the vielbein and...

How many constraints are there when the particle is moving in a plane and on three dimensional space and what are they?

Hi everyone. I have this problem which I am trying to formulate. Basically, I have the following linear constraints: p_{11} = 2 p_{22} = 5 p_{33}+2p_{12}=-1 2p_{13} =2 2p_{23} = 0 And these are for the symmetric matrix \mathbf{P} = \left( \begin{array}{ccc} p_{11} & p_{12} & p_{13}...

Hello everyone, I have a problem with 4 degrees of freedom, 2 of which are superfluous. My goal is to derive the equations of motion. I have derived the equations connecting the superfluous DoFs with the independent ones, however they are nonlinear and interlaced, which means that I cannot...

Why does a distribution function F_X have to be right continuous ? is'nt just making sure it is non-decreasing enough

1.Find a such that y=x2 - 2(x)1/2 + 1 is perpendicular to ay + 2x =2 when x=4 3.I have gotten as far as getting the slope of the normal line. I then rearranged the equation to y = (-2x+2)/ (a) and That is where I am stuck I am having a lot of trouble with these types of...

I'm trying to minimize a function over a rather complicated surface. I'm using an algorithm that takes an initial guess, finds the tangent plane at that point, minimizes using a linear programming algorithm, then (tries to) project back onto the complicated surface. More specifically, if \xi...

Question: Given a1+a2+a3+...+an=0 and a1^2+a2^2+...+an^2=1, (all real numbers) find the maximal value of a1*a2+a2*a3+...+an*a1 Thoughts so far: I've treated the expression as a combination of n variables and differentiated - when it came to putting the constraints in it got to be a...

If I have a set of Probability distributions on a product space with marginal constraints, is there any way to (how to) express the same as a linear family of PD's ( i.e. all P s.t. E_P[ f_i] =a_i for some f_i, a_i )

Homework Statement By using the Lagrange multipliers find the extrema of the following function: f(x,y)=x+y subject to the constraints: x2+y2+z2=1 y+z=12. The attempt at a solution Using lambda = 1/(2x) I got x=y-z and y=1-z plugging that into the first constraint, I got: 6y^2-6y+1=0 which...

Hello everyone, I'd like to solve overdetermined system of linear equations (in fact to fit experimental data) (like y1=C1*X11+C2*X12+...+Cm1*X1m) y2=C1*X21+C2*X22+..+Cm*X2m ... yn=C1*Xn1+C2*Xn2+...Cm*Xnm) sometimes n>>m sometimes n>~m , yi and xij are...

The speed of light is fast but far from instantaneous. What constrains the speed of light to be what it is? e.g. c = f(xi) Life is good, d

Ok, here's the question, have patience with my terrible latex skills... Homework Statement The equations of constraint of the rolling disk: dx - asin(theta)d(phi) = 0 -> 1. dy + acos(theta)d(phi) = 0 -> 2. are special cases of general linear diff-eqs of constraint of...

The discussion about the status of lorentz invariance, and as I wanted to twist it, in general, the physical and scientific basis for symmetries started to diverge in my posts so I started a new thread for this to comment more without contaminating the original thread. Thanks for the links...

MTd2 spotted this paper on arxiv and flagged it for us: Yes! I am very glad to get this one. This paper follows up on a March 2009 video seminar talk Giovanni A-C gave at Perimeter. I'll get the link. Yeah, it's easy to google: just say "Amelino Perimeter". This often works, google...

Why when doing a Lagrange Multipler with two constraints, why do you add the gradients of the two constriant funcions and set it parallel to the function to be maximized...

Homework Statement Find the maximum values of f(x,y,z)=xy+xz+yz-4xyz subject to the constraints x+y+z=1 and x,y,z>or equal 0. Homework Equations The Attempt at a Solution see attachment I found fgrad and ggrad and set fgrad equal to lambda*ggrad and used x+y+z=1 for my system of...

Homework Statement see problem 7 attachement Homework Equations The Attempt at a Solution see problem 7 attempt attachment I found the fgrad and ggrad and hgrad. I set fgrad equal to lambda*ggrad+mu*hgrad. using the two constraint equations I was able to solve for...

Suppose that you have a set of real variables {x1,x2,...,xn}. If x1 = f(x2,x3,...,xn) then this represents a constraint on all the variables. In this case, it's possible to find dx1/dxi as long as f is differentiable. But not all possible constraints among the xj are of this form. How might...

Homework Statement Consider the cylinder S in R3 defined by the equation x^2+y^2=a^2 (a). The points A=(a,0,0) \: and \: B = (a \cos{\theta}, a \sin{\theta}, b) both lie on S. Find the geodesics joining them. (b). Find 2 different extremals of the length functional joining A=(a,0,0)...

Homework Statement A solution to efficiently convert the potential energy of falling water into mechanical energy that will lift a weight is required. At the moment, I'm working to optimize my waterwheel design. A water hose with given dimensions will be placed above a platform 1 metre in...

Hi, I've been learning about the calculus of variations this term, and we just covered the transversality condition for an optimization problem with the right end point free, as well as the first necessary condition on the augmented Lagrangian for a problem with integral constraints. I'm...

let be the Lagrangian (1/2)m( \dot x ^{2} + \dot y^{2}) - \lambda (x^{2}+y^{2}-R^{2}) with 'lambda' a Lagrange multiplier , and 'R' is the radius of an sphere. basically , this would be the movement of a particle in 2-d with the constraint that the particle must move on an sphere of...

I understand that vacuum fluctuations can spring into and out of existence within a sufficiently short period of time, under the uncertainty constraint. However, I am currently a little confused over how this constraint is applied. 1) Does one use del E.del t = h bar (the commutation...

i get the idea of the laurent expansion but i get confused with the constraints and how they change the way you work with the expansion. by now you can prob. tell I am trying to get to grasp with complex analysis as a whole. for example i have this : find laurent series for :\[ f(z) =...

Prove that if a, b, r, and s are positive reals and r + s = 1, then ar bs ≤ ra + sb.

I was just wondering... if a problem involved a particle which was constrained to move from x = -a/2 to a/2 and asked you to find it's properties (not position, though), could you just "shift" the entire system from x = -a/2, a/2 to x = 0 to a? Also, let's say that a question asked for the...

m equations of semi-holonomic constraints can be put in the form: fi=(q1,q2,...,qn,\dot{q}1,...,\dot{qn}) but "commonly appears in the restricted form: \Sigmaaikdqk +aitdt = 0 (i,k,t preceded by "a" should appear in subscript and the sum is over k) I don't understand this form. what are...

1a) Determine the maximum value of f(x,y,z)=(xyz)1/3 given that x,y,z are nonnegative numbers and x+y+z=k, k a constant. 1b) Use the result in (a) to show that if x,y,z are nonnegative numbers, then (xyz)1/3 < (x+y+z)/3 Attempt: 1a) Using the Lagrange Multiplier method, I get that the...