Recent content by SilverSoldier

I'm sorry I still don't understand it and I am quite confused. Shouldn't we have ##\displaystyle\min_{\textbf{x}\in S}f\left(\textbf{x}\right)=\max_{\boldsymbol{\lambda}}\max_{\boldsymbol{\alpha}, \boldsymbol{\alpha}\geq0}\min_{\textbf{x}}L\left(\textbf{x}, \boldsymbol{\lambda}...

But it is necessary to find the minimum over values in ##S## only. How does an unconstrained minimization of ##f## over all values of ##\textbf{x}## result in its minimum over values in ##S##? :confused:

I'm reading the book Deep Learning by Ian Goodfellow, Yoshua Bengio, and Aaron Courville, and currently reading this chapter on numerical methods--specifically, the section on constrained optimization. The book states the following. Suppose we wish to minimize a function...

Suppose ##A## and ##B## are two liquids, and intermolecular forces between molecules of ##A## are of magnitude ##f_{AA}## and between molecules of ##B##, ##f_{BB}##. If ##f_{AA}>f_{BB}##, then the pure liquid ##A## is volatile than ##BB##, i.e., the tendencies that the molecules have to leave...

How are we able to simplify the above cubics to simpler equations that can be solved? How do we, for example, simplify the cubic in ##p## to obtain the usual result that ##p=\sqrt{cK_a}##?

Suppose we add a weak acid HA into pure water, so that upon addition its initial concentration is c. The following equilibria should establish in the system. $$\text{HA}+\text{H}_2\text{O}\rightleftharpoons\text{H}_3\text{O}^++\text{A}^-$$...

Suppose each side has initial length ##l##. The solution taught to me is as follows. Considering the lengths of the rods after expansion, we write $$\dfrac{3l^2}{4}\left(1+\mu\theta\right)^2+\dfrac{l^2}{4}\left(1+\alpha\theta\right)^2=l^2\left(1+\lambda\theta\right)^2$$ according to the...

Had ##x## been chosen to be the distance to a "thin line of water" as measured from a different fixed line, such as maybe the line ##AB##, then the limits of integration should vary through the distance measured to the uppermost "thin line of water" from ##AB##, and the distance measured to the...

Well, I believe you must reconsider the range of values through which the variable ##x## you chose (which is the distance that the thin layer of water is below the level of ##OP##, and not from the surface of the water itself), actually varies through, in describing the entire column of water...

When the movement of these particles is on a smooth surface, the normal force is anyway constant, right?

Thank you very much for your responses. I guess most of my doubts are now clarified. :biggrin: So in the event that the two particles are connected with a perfectly elastic string, we can use the Law of Conservation of Momentum along and perpendicular to the direction of the string, and...

What do you mean by tension required to have them move in the required manner? We do not get to decide how they move after an impulse, do we? How will they move? From what I understand, they can either move in the ##y## direction, or move in a circular path. How do we know which of these...

Mmm, do you mind telling me how? :frown: How can the tension that arises in the string be calculated, knowing only their initial and final velocities, and the impulse experience? By the way, in case it was not clear, I am not asking about the tension that arises in the string while the particle...

1. When an object attached to a fixed point with a string, is given a velocity and the string goes taut. So it says in this book (Applied Mathematics 1 by L. Bostock and S. Chandler) that when the string goes taut, the component of the velocity of the particle becomes zero in the direction...

Thank you very much for these suggestions :smile:. But I get the following with var xOffset = xVelocity*time + (output.i*Math.pow(time, 2))/2; var yOffset = yVelocity*time + (output.j*Math.pow(time, 2))/2; xVelocity += xVelocityChange; yVelocity += yVelocityChange; With xVelocity +=...