Conditions Definition and 978 Threads

The sum of the digits of a two-digit number is $12$ The number is $13$ times the tens digit. Find the number well from $3+9=12$ we can see that the number would be $39$ which is $3\times 13$ but again I had trouble knowing how to set this up In that $10t+u=12$ how do you set up $13t=$...

Let R be an integral domain. Then R is a PID if the following 2 conditions hold: 1) any 2 elements a, b \in R have a greatest common divisor which can be written as ra + sb for some r, s \in R . 2) If a_1, a_2, ... are nonzero elements of R such that a_{i+1}|a_i for all i, then there is...

"In the minutes following the explosion, protons and neutrons collided in nuclear fusion reactions to form hydrogen and helium." I found this bit of text in an article...and it got me thinking. What were the conditions the first atoms formed? Further more..what were the conditions that...

Hi, If I have two sinusoidal waves with the same frequency, wavelength and amplitude, traveling along the same line in opposite directions, the net displacement of the resulting standing wave wave is given by D(x,t) = 2a*sin(kx)*cos(ωt) the boundary conditions for standing waves on a...

Homework Statement Solve the diffusion equation: u_{xx}-\alpha^2 u_{t}=0 With the boundary and initial conditions: u(0,t)=u_{0} u(L,t)=u_{L} u(x,0=\phi(x) The Attempt at a Solution I want to solve using separation of variables... I start by assuming a solution of the form...

Hi, I'm reading through a paper and have come across what my tutor described as a 'theta function', however it seems to bear no resemblance to the actual 'theta function' I can find online. In the paper it reads: \int^1_0 dz~\theta (s-\frac{4m^2}{z}-\frac{m^2}{1-z}) And apparently this...

$$ u_{tt} + 3u_t = u_{xx}\Rightarrow \varphi\psi'' + 3\varphi\psi' = \varphi''\psi. $$ $$ u(0,t) = u(\pi,t) = 0 $$ $$ u(x,0) = 0\quad\text{and}\quad u_t(x,0) = 10 $$ \[\varphi(x) = A\cos kx + B\sin kx\\\] \begin{alignat*}{3} \psi(t) & = & C\exp\left(-\frac{3t}{2}\right)\exp\left[t\frac{\sqrt{9...

Homework Statement The steady state temperature distribution T(x,y) in a flat metal sheet obeys the partial differential equation: \displaystyle \frac{\partial^2 T}{\partial x^2}+ \frac{\partial^2 T}{\partial y^2} = 0 Seperate the variables in this equation just like in the...

μ^{2}\frac{d^{2}u}{dx^{2}}+ae^{u}=0 Boundary conditions: u(-L)=u(L)=u_{0} Solve by multiplying by \frac{du}{dx} and integrating in x I know you have to use substitution, but I keep going in circles.

Hi All, I was wondering if it is correct to say that a vanishing metric determinant is a necessary (but probably not sufficient) condition for a curvature singularity to exist at some point(s), or is one forced to construct the full Kretschmann scalar? Cheers! FD

Homework Statement If a, b and c are nonzero distinct real numbers and c > 0. The even function given by: f(x) = (ax + b)/(x + c) is constant for -c < x < c and it's value is (i) a + b; (ii) a + c; (iii) c; (iv) b; (v) a Homework Equations The Attempt at a Solution...

Laplace axisymmetric $u(a,\theta) = f(\theta)$ and $u(b,\theta) = 0$ where $a<\theta<b$. The general soln is $$ u(r,\theta) = \sum_{n=0}^{\infty}A_n r^n P_n(\cos\theta) + B_n\frac{1}{r^{n+1}}P_n(\cos\theta) $$ I am supposed to obtain $$ u(r,\theta) = \sum_{n =...

I'm not sure how to solve this: du/dt = 3 \frac{d^{2}u}{dx^{2}} These are the conditions: u(0,t)= -1 u(pi,t)= 1 u(x,0) = -cos 7x Suggestion: I should use steady state solution to get a homogeneous initial condition. Starting with separtion of variables u(x,t) = G(x)H(t) And...

Homework Statement "At 310K thermal energy kT=4.28x(10^-21). Use the equation you derived above (which I worked out to be E=(n²h²)/(8mL²) )to determine under which conditions quantum mechanics reduces to classical mechanics." The hint was that "you need to find the value of mL² for which change...

Homework Statement Using the definition of linearity to determine whether or not ech case is a linear homegeneous boundary condition: i.) Uxx(0,y)=Ux(0,y)U(0,y) ii.)Uy(x,0)=Ux(5,y) Homework Equations The Attempt at a Solution I know Uxx(0,y)=Ux(0,y)U(0,y) is not linear...

Hello, My problem is as follows: I want to generate a series of 24 dimensional random numbers to act as the starting population for a genetic algorithm. These numbers need to fully span the space which is limited by a series of nonlinear boundary conditions. The 24 dimensional vector is...

I have already solved the main portions. I have $$ T(x,t) = \sum_{n = 1}^{\infty}A_n\cos\lambda_n x\exp(-\lambda_n^2t) $$ The eigenvalues are determined by $$ \tan\lambda_n = \frac{1}{\lambda_n} $$ The initial condition is $T(x,0) =1$. For the particular case of $f(x) = 1$, numerically...

$$ \left.(\phi_n\phi_m' - \phi_m\phi_n')\right|_0^L + (\lambda_m^2 - \lambda_n^2)\int_0^L\phi_n\phi_m dx = 0 $$ where $\phi_{n,m}$ and $\lambda_{n,m}$ represent distinct modal eigenfunctions which satisfy mixed boundary conditions at $x = 0,L$ of the form \begin{alignat*}{3} a\phi(0) + b\phi'(0)...

Hello, This was an exam question which I wasn't sure how to solve: Suppose f is entire and |f(z)| \leq C(1+ |z|)^n for all z \in \mathbb{C} and for some n \in \mathbb{N}. Prove that f is a polynomial of degree less than or equal to n. I know that f can be expressed as a power series, but I'm...

Homework Statement Show that sin(z) satisfies the condition. (Stated in the title) Homework Equations The Attempt at a Solution f(z) = sin (z) = sin (x + iy) = sin x cosh y + i cos x sinh y thus, u(x,y)=sin x cosh y ... v(x,y)= cos x sinh y du/dx = cos x...

Homework Statement I'm working on some things to do with linearized gravitational radiation and I'm trying to justify the claim that in the Lorenz gauge, where \partial_{\nu}\bar{h}^{\mu\nu}=0 (1.1), we are able to impose the additional conditions A_{\alpha}^{\alpha}=0 (1.2) and...

can anyone say what is the necessary and sufficient conditions for being a tuple of real numbers ,a vector?

Homework Statement I have a general wave equation on the half line utt-c2uxx=0 u(x,0)=α(x) ut(x,0)=β(x) and the boundary condition; ut(0,t)=cηux where α is α extended as an odd function to the real line (and same for β) I have to find the d'alembert solution for x>=0; and show that in...

Hi, it is known that given two subgroups H\subset G, and K\subset G of some group G, then we have that: 1) H, K are normal subgroups of G 2) H\cap K is trivial are sufficient conditions for H and K to commute. Moreover we have that: H, K commute \Rightarrow H, K are normal. In fact...

Homework Statement we know that, every force systems can be generally replaced by a resultant force(R) and a couple(M) at a point O and the position of point O is optional. but magnitude and direction of M is dependent to this point while magnitude and direction of R is independent. In static...

Hi all, I'm doing what should be a pretty simple problem, but some theory is giving me trouble. Basically, in this problem I have a conducting sphere, surrounded by a thick insulating layer, and then vacuum outside that. I'm attempting to solve for the potential in the insulating layer by...

Hi, Say I have this pde: u_t=\alpha u_{xx} u(0,t)=\sin{x}+\sin{2x} u(L,t)=0 I know the solution for the pde below is v(x,t): v_t=\alpha v_{xx} v(0,t)=\sin{x} v(L,t)=0 And I know the solution for the pde below is w(x,t) w_t=\alpha w_{xx} w(0,t)=\sin{2x} w(L,t)=0 Would...

Hello there I'd like to know if anyone has a proof of why the condition of symmetry or antisymmetry must be followed by a FIR filter, in order for it to have a linear phase response? I've been pouring over this for an exam, and my initial question was what constitutes a linear phase...

Hi all. Let's say I want to reproduce the support conditions for a beam. The easiest one I could think of is fixed end. Like I hammer an end of the beam into the wall. This represents fixed boundary condition. Likewise can anyone point out how to reproduce Simply supported end condition in...

Hello , i am trying to implement this algorithm for 2d grid. 1) i am not sure if my calculations are correct. 2 ) i don't understand how to return my final calculation ( how will i insert to the matrix i want (the 's' in this example) the new coordinates (xup,xdow,yup,ydown)). I mean ...

The electric field in a cubical cavity of side length L with perfectly conducting walls is E_x = E_1 cos(n_1 x \pi/L) sin(n_2 y \pi/L) sin(n_3 z \pi/L) sin(\omega t) E_y = E_2 sin(n_1 x \pi/L) cos(n_2 y \pi/L) sin(n_3 z \pi/L) sin(\omega t) E_z = E_3 sin(n_1 x \pi/L) sin(n_2 y \pi/L)...

"Prove that the Bohr hydrogen atom approaches classical conditions when [. . .]" Homework Statement The problem and its solution are attached as ProblemSolution.jpg. Homework Equations E_k = chR/(n_k)^2 E_l = chR/(n_l)^2 ΔE = hc/λ hc/λ = chR[1/(n_k)^2 – 1/(n_l)^2] 1/ λ = R[1/(n_k)^2 –...

Homework Statement Determine values of a, b, c in the formula ax^2+bx^2 +c that satisfy the conditions: f(0)=0 // Limx->-1 F(x)=3 // limx->2 f(x)=6 The Attempt at a Solution 1. F(0)=0 therefore x=0 so f(0)=a(0)^2+b(0)+c so f(0)=c = 0 so c=0 2. Lim f(x) = 3, x->-1 so...

Homework Statement Solve the initial boundary value problem u_{tt}=c^2u_{xx} u(-a,t)=0,\quad u(a,t)=0,\quad u(x,0)=\sin(\omega_1 x)-b\sin(\omega_2x) where a, b, \omega_1, \omega_2 are positive constants. Homework Equations d'Alembert's solution The Attempt at a Solution...

Are there other ways of determining whether or not a function of a complex variable is analytic without using the Cauchy Riemann conditions? It seems for more complicated functions it's too difficult to decompose an arbitrary function into its real and imaginary parts, so it would be nice if...

Homework Statement Find two lines in R3 in parametric form which satisfy the following conditions. Also, find the points on the lines which achieve the closest distance. Conditions: 1. They are not parallel to any of the coordinate planes 2. They do not intersect and are not parallel 3. They...

second ODE, initial conditions are zeros at infinity! I want to know the temperature profile of phase transition layer in the interstellar medium. For stationary solution, the dimensionless differential equation I ended up with is \frac{d^2T}{dx^2} = \frac{f(T)}{T^2} - \frac{1}{T} where f(T)...

Homework Statement It's a cilinder of mass M and radius R rolling without slipping and I'm asked to find the maximum value of the static friction coefficient for the cilinder to roll without slipping. Homework Equations Non-slip conditions: \displaystyle v=\omega R \displaystyle...

Could you advise a DSP book which explains state-space representation with initial conditions. Initial conditions part is important for me. Thanks.

Long title, I know! :-p Homework Statement The heat of combustion of methane (\text{CH}_4) is 890.4 \ \text{kJ/mol} and the heat of capacity of \text{H}_2\text{O} is 75.2 \ \text{J}/\text{mol}\cdot\text{K} and that the heat capacity of \text{H}_2\text{O} is 37.7 \...

Hello everyone, I have for loop in Mathematica. After every step Mathematica has new results calculated. I want to see only some results, those who met certain conditions, not results from all steps. Say, I have 10000 steps and I don't need all 10000 results, but only those that met certain...

Not really a specific problem, but just a general question: Does anyone have any good references (preferably online) for solving E&M problems with this method? I'm using Griffith's Electrodynamics book for my class and I'm trying to get ready for a final. This is the only part I'm having...

Homework Statement not an equation as such. new to this but i wanted to know what the conditions for nuclear fission are? other than high temperatue and pressure Homework Equations "state the conditions for fusion and hence explain why it has proved difficult to maintain a sustainable...

Homework Statement Show that the following three conditions of a metric space imply that d(x, y)=d(y, x): (1) d(x, y)>=0 for all x, y in R (2) d(x, y)=0 iff x=y (3) d(x, y)=<d(x, z)+d(z, y) for all x, y, z in R (Essentially, we can deduce a reduced-form definition of a metric space...

Homework Statement Okay from what I have learned to prove that a series converges via the alternating test, you must prove the following conditions Homework Equations 1) an > 0 2) lim an (n--> infinity) = 0 and 3) a(n+1) < an The Attempt at a Solution However recently I've been encountering...

\begin{align} \varphi''+\lambda\varphi &= 0, & \quad 0< x < L\\ \varphi'(0) &= 0 &\\ \varphi'(L)+h\varphi(L) &=0, & \quad h\in\mathbb{R} \end{align} $$ \varphi = A\cos x\sqrt{\lambda} + B\frac{\sin x\sqrt{\lambda}}{\sqrt{\lambda}} $$ Since $\varphi'(0) = 0$, $\varphi = A\cos x\sqrt{\lambda}$...

Tonight on 20/20 10PM EST/9PM CST. I know some people would like to watch this.

Well i know real gases behave as ideal gas (almost) when pressure is low and temperature is high. I want to understand this - When pressure is low attractive forces in the gas moelcules will be stronger(as compared to high pressure) but the fast movement due to high temperature compensates it...

Homework Statement I have boundary conditions on my heat equation ## \dot{T}(x,t) = T''(x,t) ## ## T(0,t) = T(L,t), ## ## \frac{\partial T(0,t)}{\partial x} = \frac{\partial T(L,t)}{\partial x} ## Then at ## T= 0## ## T(x,0) = 1 ## for ## 0<x<L/4 ## ## T(x,0) = 0 ## for ## L/4<x<L ##...

If only some stars become black holes what sets them apart from other stars, is it just size?