I've been reading up on Fourier Transform Infrared Spectroscopy and the Michelson Interferometer. My main sources are "Principles of Instrumental Analysis" by Skoog etc and Fourier - Transform Infrared Spectrometry by Griffiths and Haseth.
I believe I understand the theoretical principles...
So I'm reading "An Introduction to Wavelet Analysis" by David F. Walnut and it's saying that the following sequence
" (x^n)_{n\in \mathbb{N}} converges uniformly to zero on [-\alpha, \alpha] for all 0 < \alpha < 1 but does not converge uniformly to zero on (-1, 1) "
My problem is that isn't...
So I'm reading the Example on page 161 of Differential Equations, Dynamical Systems and an Introduction to Chaos by Hirsh, Smale, and Devaney.
I'm not understanding everything.
So given the system
x' = x + y^2
y' = -y
we see this is non-linear. I get it that near the origin, y^2 tends to...
So z'(t) = x'(t) + i y'(t), I just want to make sure that what I'm going to do is OK. I'm trying to solve z' = az where a = \alpha + i \beta
z'(t) = az(t) \Rightarrow x'(t) + iy'(t) = a(x(t) + iy(t)) \Rightarrow x'(t) = a x(t), y'(t) = a y(t)
If you could give me a justification that'll...
Homework Statement
Consider the first-order non-autonomous equation ##x' = p(t) x##, where ##p(t) ## is differentiable and periodic with period ##T##. Prove that all solutions of this equation are period with period ##T## if and only if ##\int_0^T p(s) ds = 0##.
Homework Equations
The...
Homework Statement
Let f be a non-negative measurable function. Prove that
\lim _{n \rightarrow \infty} \int (f \wedge n) \rightarrow \int f.
The Attempt at a Solution
I feel like I'm supposed to use the monotone convergence theorem.
I don't know if I'm on the right track but I created...
Homework Statement
Define
f_n : \mathbb{R} \rightarrow \mathbb{R} by
f_n(x) = \left( x^2 + \dfrac{1}{n} \right)^{\frac{1}{2}}
Show that f_n(x) \rightarrow |x| converges uniformly on compact subsets of \mathbb{R}
Show that the convergence is uniform in all of \mathbb{R}...
So when we have an open cover of a set X means we have a collection of sets \{ E_\alpha\}_{\alpha \in I}
such that X \subset \bigcup_{\alpha \in I} E_\alpha .
My question comes from measure theory, on the question of finite \sigma -measures,
The definition I'm readying says \mu is \sigma...
Homework Statement
Find a set X such that \mathcal{A}_1 \text{ and } \mathcal{A}_2 are \sigma-algebras where both \mathcal{A}_1 \text{ and } \mathcal{A}_2 consists of subsets of X. We want to show that there exists such a collection such that \mathcal{A}_1 \cup \mathcal{A}_2 is not a...
Hello,
I'm reading "Complex Made Simple" by David Ullrich.
He has these notation for disks
D(z_0,r) = \left\{ z \in \mathbb{C}: |z-z_0|< r \right\}
\bar{D}(z_0,r) = \left\{ z \in \mathbb{C} : |z - z_0| \leq r \right\}
I understand that these sets are to be the open and closed disks...
Homework Statement
If \phi \in \mathcal{M} (group of all linear fractional transformations or Mobius Transformations has three fixed points, then it must be the identity. (The proof should exploit the fact that \mathcal{M} is a group.
The Attempt at a Solution
Hi all,
So...
Homework Statement
Define f : \mathbb{C} \rightarrow \mathbb{C} by
f(z) = \left
\{
\begin{array}{11}
|z|^2 \sin (\frac{1}{|z|}), \mbox{when $z \ne 0$}, \\
0, \mbox{when z = 0} .
\end{array}
\right.
Show that f is complex-differentiable at the origin although the...
Homework Statement
Consider the nonlinnear diffusion problem
u_t - (u_x)^2 + uu_{xx} = 0, x \in \mathbb{R} , t >0
with the constraint and boundary conditions
\int_{\mathbb{R}} u(x,t)=1, u(\pm \inf, t)=0
Investigate the existence of scaling invariant solutions for the equation...
Homework Statement
Use the method of characteristics to solve the problem:
-xu_x + yu_y = 2xyu and u(x,x)=x
Homework Equations
The Attempt at a Solution
let x=x(t), y=y(t), u=u(x(t),y(t))
so
\frac{du}{dt} = \frac{∂u}{∂x} \frac{dx}{dt} + \frac{∂u}{∂y} \frac{dy}{dt}
and...
I'm reading Lee's Introduction to Smooth Manifolds and I have a question on the definition of a multilinear function
Suppose V_{1},...,V_{k} and W are vector spaces. A map F:V_{1} \times ... \times V_{k} \rightarrow W is said to be multilinear if it is linear as a function of each variable...
Homework Statement
\frac{dU}{dz} = V, \frac{dV}{dz}=k+cU-6U^{2} c \in ℝ
Find the fixed points of the system (these are solutions U=U*, V=V* where U*,V* \in ℝ) and determine the value of k so that the origin is a fixed point of the system
Homework Equations
The Attempt at a...
Homework Statement
let \ell^{2} denote the space of sequences of real numbers \left\{a_{n}\right\}^{\infty}_{1}
such that
\sum_{1 \leq n < \infty } a_{n}^{2} < \infty
a) Verify that \left\langle \left\{a_{n}\right\}^{\infty}_{1}, \left\{b_{n}\right\}^{\infty}_{1} \right\rangle =...
Homework Statement
Show that the image of the curve Let β: (-π,π) → ℝ2 be given by β(t) = (sin2t, sint)
is not an embedded submanifold of ℝ2
Homework Equations
The Attempt at a Solution
So I'm not too great with the topology. I do see that β'(t) = (2cos2t, cost) ≠ 0 for all t. So β is a...
Homework Statement
Consider the map \Phi : ℝ4 \rightarrow ℝ2
defined by \Phi (x,y,s,t)=(x2+y, yx2+y2+s2+t2+y)
show that (0,1) is a regular value of \Phi and that the level set \Phi^{-1} is diffeomorphic to S2 (unit sphere)
Homework Equations
The Attempt at a Solution
So I...
Let U be a subset of ℝn be an open subset and let f:U→ℝk be a continuous function.
the graph of f is the subset ℝn × ℝk defined by
G(f) = {(x,y) in ℝn × ℝk : x in U and y=f(x)}
with the subspace topology
so I'm really just trying to understand that last part of this definition...
There's this theorem:
A linear map T: V→W is one-to-one iff Ker(T) = 0
I'm wondering if there's an analog for showing that T is onto? If so could you provide a proof?
I'm thinking it has something to do with the rank(T)...
So let ℝ^{n}_{a}={(a,v) : a \in ℝ^{n}, v \in ℝ^{n}}
so any geometric tangent vector, which is an element of ℝ^{n}_{a} yields a map
Dv|af = Dvf(a) = \frac{d}{dt}|_{t=0}f(a+tv)
this operation is linear over ℝ and satisfies the product rule
Dv|a(fg) = f(a)Dvg + g(a)Dvf
if v|a =...
Homework Statement
If M and N are smooth manifolds, then T(MxN) is diffeomorphic to TM x TN
Homework Equations
The Attempt at a Solution
So I'm here
let ((p,q),v) \in T(MxN)
then p \in M and q \in N and v \in T(p,q)(MxN).
so T(p,q)(MxN) v = \sum_{i=1}^{m+n}...
I'm reading the second edition of John M. Lee's Introduction to Smooth Manifolds and he has a proposition that I'd like to understand better
Let M, N, and P be smooth manifolds with or without boundary, let F:M→N and G:N→P be smooth maps and let p\inM
Proposition: TpF : TpM → TF(p) is...
Homework Statement
Consider the IVP
\frac{dy}{dt} = t2 + y2, y(0)=(0)
and let B be the rectangle [0,a] x [-b,b]
a) the solution to this problem exists for
0≤t≤min{a, \frac{b}{a2+b2}
b) that min{a,\frac{1}{2}a} is largest when a=\frac{1}{\sqrt{2}}
c) Deduce an interval 0≤t≤α on which the...
Homework Statement
Find a solution of the IVP
\frac{dy}{dt} = t(1-y2)\frac{1}{2} and y(0)=0 (*)
other than y(t) = 1. Does this violate the uniqueness part of the Existence/Uniqueness Theorem. Explain.
Homework Equations
Initial Value Problem \frac{dy}{dt}=f(t,y) y(t0)=y0 has a...
Homework Statement
Let X be the set of all points (x,y)\inℝ2 such that y=±1, and let M be the quotient of X by the equivalence relation generated by (x,-1)~(x,1) for all x≠0. Show that M is locally Euclidean and second-countable, but not Hausdorff.
Homework Equations
The Attempt at...
on page 261 of this paper by J. Vermeer (http://www.math.technion.ac.il/iic/e..._pp258-283.pdf [Broken]) he writes
The following assertions are equivalent.
a) A is similar to a Hermitian matrix
b) A is similar to a Hermitian matrix via a Hermitian, positive definite matrix
c) A is similar...
Homework Statement
If A≥0 and Ak>0 for some k≥1, show that A has a positive eigenvector.
Homework Equations
The Attempt at a Solution
A is nxn
Well from a previous problem we know that the spectral radius ρ(A)>0
We also know that if A≥0, then ρ(A) is an eigenvalue of A and...
I need some help understanding the following definition:
Definition: Let A\inMn(ℂ) the complex vector space
C(A)={X\inMn(ℂ) : XA=AX}
For A\inMn(ℂ) which is similar to A* we define the complex vector spaces:
C(A,A*)={S\inMn(ℂ) : SA=A*S}
H(A,A*)={H\inMn(ℂ): H is Hermitian and...