Well, what I was trying to say was that the upper sum cannot equal 0 because for any interval [a,b] M will not be 0 because there is always an irrational greater than 0 no matter how small the partition [Xo, X1] gets. Because M will never be 0 the inf of the Upper Darboux Sums will never be 0...
Let f:[0,1] be defined as f(x)= 0 for x rational, f(x)=x for x irrational
Show f is not integrable
m=inf(f(x) on [Xi-1, Xi])
M=sup(f(x) on [Xi-1, Xi])
Okay so my argument goes like this:
I need to show that the Upper integral of f does not equal the lower integral of f
Because...
So I've been trying to figure this out. The question is:
If the limit x->infinity of Xn=Xo
Show that, by definition, limit x->infinity sqrt(Xn)=sqrt(Xo)
I'm pretty sure I need to use the epsilon definition.
I worked on it with someone else and we think that what we have to show is the...
hi, I have two equations that I have used the Solve function in Mathematica to solve for A in both equations
What I am having trouble with is trying to equate the results and solving for another variable J automatically
Basically this is what I want to do:
Solve[Eqn1, A]
It gives {{A ->...
This may be a dumb question, but I jsut want to make sure I understand this correctly.
For R_{1}, R_{2}, ..., R_{n}
R_{1} \oplus R_{2} \oplus, ..., R_{n}=(a_{1},a_{2},...,a_{n})|a_{i} \in R_{i}
does this mean that a ring which is a direct sum of other rings is composed of specific elements...
I'm trying to solve this equation:
Ux + Uy + U = e^-(x+y) with the initial condition that U(x,0)=0
I played around and and quickly found that U = -e^-(x+y) solves the equation, but does not hold for the initial condition. For the initial condition to hold, I think there needs to be some...
The problem statement is:
Solve the Neumann problem for the wave equation on the half line 0<x<infinity.
Here is what I have
U_{tt}=c^{2}U_{xx}
Initial conditions
U(x,0)=\phi(x)
U_{t}(x,0)=\psi(x)
Neumann BC
U_{x}(0,t)=0
So I extend \phi(x) and \psi(x) evenly and get...
I'm following an algorithm my teacher gave us and I'm trying to understand it...
I'm trying to solve this PDE 2Ux-Uy+5U=10 with U(x,0)=0
First I need to solve the homogeneous equation.
So I set up the relation:
V(y)=U(2y+c, y)
to solve 2Ux-Uy=0
where the characteristic equation is y=1x/2...
Consider heat flow in a long circular cylinder where the temperature depends only on t and on the distance r to the axis of the cylinder. Here r=\sqrt{x^{2}+y^{2}} is the cylindrical coordinate. From the three dimensional heat equation derive the equation u_{t}=k(u_{rr}+\frac{u_{r}}{r}).
My...
Okay so I get your method here and I am trying to apply it to this one (23)(13) but I am not getting the answer the book has which is (123)
I set it up like this
(23) (13)
123 123
132 321
then
so 1 goes to 3 then 3 goes to 2, 2 goes to 2 then 2 goes to 3, 3 goes to 1 and 1 goes...
I am having trouble understanding this example:
Let G=S_3 and H={(1),(13)}. Then the left cosets of H in G are
(1)H=H
(12)H={(12), (12)(13)}={(12),(132)}=(132)H
I cannot figure out how to produce this relation:
(12)H={(12), (12)(13)}={(12),(132)}=(132)H
I understand (12)H={(12), (12)(13)}...
To give a bit more context.
I was trying to solve the partial differential equation: 3U_{y}+U_{xy}=0 with the hint, let V=U_y
substituting we have 3V+V_{x}=0
then -3=\frac{V_{x}}{V}
I didn't really know how to continue from here so I just played around and figured out that V=e^{3x} and...
So in doing a homework problem I have convinced my self that \int\frac{V_{x}}{V}=ln(V) which I vaguely remember learning in class, but I'm having trouble deriving it. Can someone help me out?
I see what you mean by the difference in functions, however I feel like they would have written it the way you did, {1, sin2x, cos2x}, if that's what they meant? They way I see it, the function in the question represents all functions of that form which is why they have function plural...
Well to be honest I have forgotten a lot of this stuff.
I'm looking at the function
(c_{1}+c_{2}sin^{2}x+c_{3}cos^2{x})
with c1, c2, and c3, as distinct constants and that all functions of this form can be combined to create another function this form. This seems to be the wrong way to think...
Show that the functions (c_{1}+c_{2}sin^{2}x+c_{3}cos^2{x}) form a vector space. Find a basis of it. What is its dimension?
My answer is that it's a vector space because:
(c_{1}+c_{2}sin^{2}x+c_{3}cos^2{x})+(c'_{1}+c'_{2}sin^{2}x+c'_{3}cos^2{x})...
Is this linear homogeneous, linear inhomogeneous etc...
u_{t}-u_{xx}+xu=0
From that first one I get this
\frac{u_{t}-u_{xx}}{u}=-x
which I'm not sure is linear.
Edit:
Similar questions involve the following equations:
iu_{t}-u_{xx}+\frac{u}{x}=0
and
u_{x}+e^{y}u_{y}=0
Another Edit:
I...
I have two proofs that I am uneasy about and one I'm having trouble with so hopefully I can figure out where I'm going wrong if I am. Ignore the weird numbers, its to help me organize the problems.
14) Let G be a group with the following property: Whenever a, b and c belong to G and ab=ca, then...
Find integers s and t such that 1 = 7*s + 11*t. Show that s and t are not unique.
I can find numbers that satisfy this question, t=2, s=-3 and t=-5, s=8, that show s and t are not unique, but this doesn't seem to be rigorous and I'm not sure where to start with proving this.
To make it work for p_{1} I can just use p_{1} instead of p_{n}, I guess. I don't know if that helps me see what to do...
I was thinking of writing something like what I had, but instead of p_{n} on the right I would have p_{x} where x\in{1,2,...,n}, but that seems weird.
Let p_{1}, p_{2},...,p_{n} be primes. Show that p_{1} p_{2}...p_{n}+1 is divisible by none of these primes.
Let p_{1}, p_{2},...,p_{n} be primes
Let k \in N
Assume p_{1}p_{2}...p_{n}+1=kp_{n}
\frac{p_{1}p_{2}...p_{n}}{p_{n}}+\frac{1}{p_{n}}=k
p_{1}p_{2}...p_{n-1}+\frac{1}{p_{n}}=k
This is a...
I'm following the answer to a problem and I see this step which I am unsure about:
F[k]=\frac{1}{2}\int^{1}_{-1}|x|e^{-\pi i k x}dx
F[k]=\frac{1}{2}\int^{1}_{-1}|x|cos(\pi k x)dx
For k equal to all integers.
Shouldn't the conversion from the exponential be cos(-\pi k x)
Okay so I have the following problems to figure out by Monday morning. I'm very behind so I am going to spend some time trying to work them out on my own tonight and tomorrow so don't think I am asking for the specific answers, but I thought I would post this here so when I have some trouble...
I am having trouble understanding this.
If it is even and 2-periodic are you saying that it would be basically the graph of |x| on (-1,1)?
for odd just the graph of x on (-1,1)?
What confuses me if this is right is that it says in the question that the interval is (0,1)
Also can you confirm...
Let f(x):= x when 0<x<1. Find the Fourier series for f if:
a) f is 1 periodic
b) f is even and 2 periodic
c) f is odd and 2 periodic.
I am very lost and behind. I'm reading through my notes and book and hopefully will be able to to this soon, but can anyone give me a hint or just explain how...
I feel a bit dumb, but could someone help me see this:
G(s):= \int_{-\infty}^{\infty}f(-x)e^{-2\pi isx}dx = \int_{-\infty}^{\infty}f(u)e^{-2\pi i(-s)u}du = F(-s)