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    Michelson Interferometer IR Lab Question

    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...
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    Uniform Convergence question

    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...
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    Equilibria in Nonlinear Systems

    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...
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    Solving z'(t) = az(t)

    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...
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    Non-autonomous diff eq and periodicity

    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...
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    Monotone Convergence Theorem

    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...
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    Uniform Convergence of a Sequence of Functions

    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}...
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    Question on Definition of Cover of a Set

    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...
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    Question on Sigma Algebras

    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...
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    Notation on Disks

    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...
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    Complex Analysis and Mobius Transformation.

    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...
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    Complex Analysis, Complex Differentiable Question

    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...
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    Scaling Invariant, Non-Linear PDE

    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...
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    Method of Characteristics

    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...
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    Multilinear Algebra Definition

    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...
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    Dynamical System

    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...
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    Hilbert Space

    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 =...
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    Topology, lemiscate not an embedded submanifold

    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...
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    Regular Values (Introduction to Smooth Manifolds)

    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...
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    Graphs of Continuous Functions and the Subspace Topology

    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...
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    Proving a Linear Transformation is Onto

    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)...
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    Geometric Tangent Vector

    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 =...
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    Tangent Bundles, T(MxN) is Diffeomorphic to TM x TN

    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}...
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    Properties of Differentials, Smooth Manifolds.

    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...
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    Finding Intervals of Solutions to ODE's

    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...
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    Lipschitz Condition, Uniqueness and Existence of ODE

    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...
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    Topology, line with two origins

    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...
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    Positive Definite Matricies

    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...
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    Non-Negative Matrices

    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...
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    Kern(T) not the Ker(T)

    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...
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