Conditions Definition and 1000 Threads

Other than for null-homotopic maps, which continuous maps defined on ##D^1 \rightarrow D^1## (Open disk)extend continuously to maps ##B^1 \rightarrow B^1 ## ,(##B^1## the closed disk) which maps can be extended in opposite direction, i.e., continuous maps ## f: S^1 \rightarrow S^1 ## that...

The book of Balanis solves the field patterns from the potential functions. Let say for TE modes, it is: F_z(\rho, \phi, z) = A_{mn} J_m(\beta_{\rho}\rho) [C_2 \cos(m\phi) + D_2 \sin(m\phi)] e^{-j\beta_z z} There is no mention of how to solve for the constant A_{mn} . Then, from a paper...

Hi everyone! This is the first time I'm posting on any forum and I'm still rather unsure of how to format so I'm sorry if it seems wonky. I'll try my best to keep the important stuff consistent! I am working on infinite square well problems, and in the example problem: V(x) = 0 if: 0 ≤ x ≤ a...

Hello, I am trying to solve a differential equation corresponding to a visco-elastic material model consisting of 5 units of springs and dashpots connected in parallel as can be seen in the image below. I am able to come-up with a single fifth order ODE, however I am struggling to find the...

I am operating via finite differences. Say for example, I have this pipe that contains a fluid. I have the boundary condition at x = x1: k is the effective thermal conductivity of the fluid, T is the temperature of the fluid at any point x, hw is the wall heat transfer coefficient, and Tw is...

So this question is a math question having to do with me calculating the rate of population growth starting from a population of 100,000. I have already gotten the first 3 steps done (sex ratio, ratio of cycle time, and pregnancy ratio) after a week among those in the fertile timeframe...

Do we know? Do we have any idea? There seems to be 2 schools of thought. 1. The initial conditions can be almost any value with universes inflating with different laws of physics. This is the multiverse and string theory 10^500 false vacua view. 2. The initial conditions are more restrained...

Summary: Questions about the Multiverse hypothesis and the 'No boundary' conditions approach in cosmology I have some questions about James Hartle and Stephen Hawking's 'No-boundary' proposal: - In their approach multiple histories would exist. These histories could yield universes with...

I have the following PDE I wish to solve: \frac{\partial u}{\partial t}=D\frac{\partial^{2}u}{\partial x^{2}} With the following boundary conditions: \frac{\partial u}{\partial x}(t,1)+u(t,1)=f(t),\quad u(t,0)=0 Now, I wish to do this via the Crank-Nicholson method and I would naively...

I have the formula for amplitude ##A=\sqrt (x_0^2 + \frac{\dot x_0 ^2}{\omega^2})##. But ##x_0## and ##\dot x_0## refers to the initial conditions, and the information that I'm given is not related to the initial conditions, or at least I'm not told so.

Here’s the metric: $$ds^2 = -dt^2+dl^2+r^2(l)d\Omega^2$$where ##r(l)## is minimum at ##l=0## with ##r(0)=r_0## and ##r## approaching ##|l|## asymptotically as ##l## approaches ##\pm \infty## Part a of the problem seemed pretty straightforward and intuitive, but part b asks which energy...

Consider the following linear first-order PDE, Find the solution φ(x,y) by choosing a suitable boundary condition for the case f(x,y)=y and g(x,y)=x. --------------------------------------------------------------------------- The equation above is the PDE I have to solve and I denoted the...

71.15 Two vectors $\vec{A}$ and $\vec{B}$ lie in xy plane. Under what conditions does the ratio $\vec{A}/\vec{B}$ equal $\vec{A_x}/\vec{B_x}$? Sorry but I had a hard time envisioning what this would be? also thot I posted this earlier but I can't find it

PS: This is not an assignment, this is more of a brain exercise. I intend to apply a general derivative boundary condition f(x,y). While I know that the boxed formulation is correct, I have no idea how to acquire the same formulation if I come from the general natural boundary condition...

Required Equations Note: Includes inequalities. [1] H(s) = 0, and [2] H’(s) = 0? [3] [4] H = a’/a [5] a(t0) = a0 =1 [6] H0 = H(t0) [7] ΩR = 0 [8] Ωk<0, Ωk = -K, K > 0 [9] ΩM>1, ΩM = M, M > 1 [10] ΩΛ = 1-Ωk- ΩM = 1+K-M > 0 → M-1 < K Solution Note: I have assigned this problem to myself...

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Hi everyone, Someone posted this hypothetical problem on a facebook group and I am wondering what your thoughts are. The issue is that Snell's law does not seem to hold when applied to the transmitted wave (calculated using the electromagnetic interface conditions.) Here is an example...

Premise: everything that follows is done in the frequency domain. Boundary conditions If there are superficial currents (electric and magnetic) impressed on the boundary between two media, we have these discontinuities for the tangential components of the fields...

Hello! I have the following question: Let $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$. Is there any class of class of functions and some kind of "growth conditions" such that bounds like below can be established: \begin{equation} ||f(x)||\geq g\left( \text{dist}(x,\mathcal{X})\right)...

In the Classical Dynamics of Particles and Systems book, 5th Edition, by Stephen T. Thornton and Jerry B. Marion, page 220, the author derived Equation (6.67) from Equation (6.66) which is the following: Equation (6.67): $$\left(\frac{\partial f}{\partial y} − \ \frac{d}{dx}\frac{\partial...

I'm not quite sure if my problem is considered a calculus problem or a statistics problem, but I believe it to be a statistics related problem. Below is a screenshot of what I'm dealing with. For a) I expressed f(t) in terms of parameters p and u, and I got: $$f(t)=\frac{-u \cdot a + u \cdot...

Are there theorems for sufficient and necessary conditions for the vanishing of Berry and/or Wilzeck-Zee phases for a given quantum mechanical system?

Hi, just a question regarding neumann conditions, I seem to have forgotten these things already. I think this question is answerable by a yes or a no. So given the 2D heat equation, If I assign a neumann condition at say, x = 0; Does it still follow that at the derivative of t, the...

If a 3×3 matrix A produces 3 linearly independent eigenvectors then we can write them columnwise in a matrix P(non singular). Then the matrix D = P_inv*A*P is diagonal. Now for this I need to show that different eigenvalues of a matrix produce linearly independent eigenvectors. A*x = c1x A*y...

So quantum states collapse when observed, ie they are interacted with, if one had an uncharged massive particle, in a true vacuum (yes, this assumption is egregious, not your usual "ignore air resistance"), and the state of the aforementioned particle collapsed, would that imply gravitons did...

Hi everyone, in the attached file I tried to find the transmitted and the reflected coefficients. I ran into trouble applying the boundary conditions to the linear components of the electric field. Check the outlined boxes and see if they make sense. Thanks

Vo(S) = [ N(s)Vi(s) + (- s2 + s - 2) ] / s3 + s2 + 1 ; can ignore (-s^2 + s - 2). From relevant equations: Vo(S) = [N(s)*Vi(s)]/(s^3 + s^2 + 1); -> (d3Vo(t)/dt3) + (d2Vo(t)/dt2) + Vo(t) = N(t)(dvi)/dt L[vi(t)] = t to s domain: [s3Vo(s) - s2Vo(0-) - SV'o(0-) - Vo''(0-)]Vo(s) + s2 - SVo -...

Homework Statement [/B] I have a metal disc adhesively bonded at its edges to a piezoelectric ring. The piezoelectric ring vibrates radially which leads to the plate vibrating transversely. I am looking to work out the resonant frequency of the metal disc which I believe will depend on the...

Hi! I have the following sequence $$(x_{n})_{n\geq 1}, \ x_{n}=ac+(a+ab)c^{2}+...+(a+ab+...+ab^{n})c^{n+1}$$ Also I know that $a,b,c\in \mathbb{R}$ and $|c|<1,\ b\neq 1, \ |bc|<1$ I need to find the limit of $x_{n}$. My attempt is in the picture.The result should be $\frac{ac}{(1-bc)(1-c)}$ I...

Homework Statement We have an incident electric field, and there are two cases: 1) the field is polasised perpendicularly to the incidence plane (TE) 2) polarised in the plane (TM) Here I must be able to correctly apply the limit conditions, to find the Fresnel formulas that give the...

I'm studying for an exam (so this is existential) and I never really got how to write the limit conditions for an interface. In my problem, there is an incident electric field, and there are two cases: 1) the field is polasised perpendicularly to the incidence plane (TE) 2) polarised in the...

Homework Statement I am trying to fill in the gaps of a calculation (computing the deflection potential ##\psi##) in this paper: http://adsabs.harvard.edu/abs/1994A%26A...284..285K We have the Poisson equation: ##\frac{1}{x}\frac{\partial}{\partial x} \left( x \frac{\partial \psi}{\partial...

Hi,how do I go about answering the attached question? I know that for a matrix to have no solution, there needs to be a contradiction in some row. Unique solutions is when m* ${x}_{3}$ =c , where m* ${x}_{3}$ $\ne$ 0. One way I tried was if a=0, then from row (1) : b* ${x}_{3}$ =2...

Hi everyone! I have to solve a problem using Poisson's equation. There are two parallel infinite conductor planes in vacuum. The distance between them is d and they are both kept at a potential V=0. Between them there is a uniform volume density charge \rho_0>0 infinite along the directions...

I am planning a physical experiement under vacuum conditions. In this experiement, I want to detect ionizing rays, especially as broad band as possible, for instance including alpha radiation. Typically geiger tubes are filled with a certain gas, that gets ionized by the radiation to measure...

Suppose that we have a some rotating object (lets say a wheel with radius R). Let's observe this problem from some reference frame in which center of mass translates with some velocity v and rotates with angular velocity ω. I know that condition for rolling without slipping is v=ωR (point at...

Are there any conditions under which glass cracks spontaneously? Just now, my mother's rear windshield cracked spontaneously. The story according to her: Slowly approaching a stop sign. No busy traffic. No trees/animals in sight. Few people. Suddenly, the rear windshield cracked with a loud boom...

Homework Statement Equation of the circle passing through the point (1,2) and (3,4) and touching the line 3x+y-3=0 is? Homework Equations x^2+y^2+2gx+2fy+c=0...(1) (-g,-f)=center of circle sqrt(g^2+f^2-c)=radius...(2) The Attempt at a Solution Putting (1,2) and (3,4) in equation 1 we get...

In the textbook (attached image) it says that the boundary condition is V=0 at r=R. This creates a correlation that ##B_l=-A_l R^{2l+1}## but the potential at any boundary is continuous so when we take this account, we get. ##B_l=A_l R^{2l+1}## These two clearly contradict each other. I'd...

Homework Statement Consider a classic wedge and block system, (block on top of wedge(inclination theta)). there is friction between the block and wedge (not enough to prevent block from sliding). All other surfaces are smooth. For the motion that follows after releasing the block from rest, is...

What are the conditions for applicability of u-substitution, i.e. when does it not work? Note that I'm not asking when is it a bad idea (that won't get you any closer to evaluating the integral), but are there any conditions that cause u-sub to yield wrong answers? I started running into what I...

Homework Statement Let ##G## be a finite group and ##m## a positive integer which is relatively prime to ##|G|##. If ##b\in G## and ##a^mb=ba^m## for all ##a\in G##, show that ##b## is in the center of ##G##. Homework EquationsThe Attempt at a Solution Let ##|G| = n## and ##b\in G##. Note that...

Homework Statement Let ##b_1\in \mathbb{R}## be given and ##n=1,2,\dots## let $$b_{n+1} := \frac{1+b_n^2}{2}.$$ Define the set $$B := \{b_1\in\mathbb{R} \mid \lim_{n\to\infty}b_n \text{ converges}\}$$ Identify the set ##B##. Homework EquationsThe Attempt at a Solution I claim that ##B =...

Hi all! I need to give a presentation on a problem in class and @countryboy helped me to figure out most of it, I only have one remaining question. Here it goes: Consider the system dr/dt = -j, dj/dt = r , where r (t) represents Romeo’s love (positive values) or hate (negative values) for...

Hi, initially I cut the photo off the book called Nonlinear-and-Mixed-Integer-Optimization-Fundamentals-and-Applications-Topics-in-Chemical-Engineering and page 120. My question is: in the GREEN box it says we have to use KKT gradient conditions with respect to a_i and as a result of that we...

We are using griffith's 4 edition in my electromagnetic course atm. and there's something I just don't understand about boundary conditions. It says that if we have a surface charge, and we put a pillbox on it, in such a way that half of it extends under the surface charge, and the other half...

How would you solve for the Amplitude(A) and Phase Constant(ø) of a spring undergoing simple harmonic motion given the following boundary conditions: (x1,t1)=(0.01, 0) (x2,t2)=(0.04, 5) f=13Hz x values are given in relation to the equilibrium point. Equation of Motion for a spring undergoing...

I don't understand from Peskin when can I use Ward Identity? I mean I can see that this identity isn't always valid to use, but when it is? Take for example equation (16.10) page 508 of Peskin's and Schroeder's.

We have a cube on an inclined plane. The tipping condition is the presence of an unbalanced torque relative to the center of mass (contributing forces are: the normal force and the force of friction). However, is this conditions equivalent to the previous one: The line of action of the force of...

Hi there. I am solving an under-determined system of equations. The solver I've written seems to work fine, but I would like to have some control on the solution I obtain. From the infinite set of solutions of my system, I would like to use only those that fulfill some conditions. One of those...