Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.

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Non-dimensionalise

A body of constant mass is thrown vertically upwards from the ground. It can be shown that the appropriate non-dimensional differential equation for the height $x(t;u)$, reached at time $t\geq0$ is ...
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2answers
26 views

How can I prove this function is not continuous for every point other than 0?

Define $g:[0,1]\rightarrow\mathbb R$ by $g(x)=\sqrt{x}$ if $x$ is rational and $g(x)=0$ if x is irrational. Prove that $g$ is continuous at $x=0$, but is not continuous at any other value of $x$. I ...
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22 views

Criteria to be in weak $L^{p}$ space

Let $X$ be a $\sigma$-finite measure space. Let $f : X \rightarrow \mathbb{C}$ be a measurable function and $1 < p < \infty$. Suppose for $f$ there is a constant $C$ such that ...
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1answer
19 views

Radius of convergence | ratio test

I need to find the radius of convergence of $\Sigma n^3z^n$ I want to use the ratio test because it would be simpler than the root test. If $C_n=n^3$ then $| \dfrac {C_{n+1}}{C_n}| > 1$ because ...
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26 views

Bounded linear functionals on $L^\infty$.

I am looking at a practice final and I am a bit confused by this statement I am trying to prove: "There is a nonzero bounded linear functional on $L^\infty[0,1]$ which vanishes on the subspace ...
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2answers
151 views

Intuition Behind Maximum Principle (Complex Analysis)

Let $D$ be an open set in the complex plane and $f(z)$ be a non-constant holomorphic function on D. Then $|f(z)|$ has no local maximum on D. I can follow the proof fine - usually if I don't ...
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38 views

a question about convergence of sequecce!I have tried cauchy method, but it doesn't work

suppose $a_n>0$,and$\sum_{i=0}^\infty a_i$ is convergent,so we need to prove $\sum_{n=1}^\infty{ {1\over n}(a_n+a_{n+1}+\cdots+a_{2n})}$ is also convergent! I have tried cauchy method, but maybe ...
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48 views

Calculus and infinitesimals

In the definition of reimann integral, why do we put a 'dx' inside the integral sign when practically it serves no purpose except maybe telling what variable you are talking about. Then in some ...
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1answer
13 views

Compostion of functions in taylor series

I was attempting to help answer a question, but I am curious about the following. Suppose we have an analytic function $f(z)\equiv\sum\limits_{k=0}^{\infty} a_k(z-z_0)^k $ on some suitable disk of ...
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1answer
9 views

Defining a region as a data structure

Is there a way for one to define a curve or region (such as a closed, 2-d disk) as a data structure into the computer, and make an algorithm which detects if a point is a boundary point, limit points, ...
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1answer
19 views

Show that span is separable [on hold]

Let $X$ be a n.v.s and $A\subset X$. Show that if $A$ is enumerable then $\overline{ \text{span}\{A\}}$ is separable
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17 views

Pointwise convergence of arctan(nx)

Question 6 section 8.1 of Introduction to real analysis by Bartle and Sherbert. Show that lim(Arctan nx) = (pi/2)sgn x for x in R, x>=0. I have a final coming up and I've started doing some of the ...
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16 views

Real analysis problem about open set and neighborhood [on hold]

Prove that If U is open set in X if and only if $\exists V\subset \mathbb{R}^{n},U=V\cap X$ (V to be open set in $\mathbb{R}^{n}$ ) I need your help. Thank you for reading
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20 views

Intermediate value of the derivative.

Hi all what would the best way be to approach this question? I tried using the hint but I can't seem to formulate an answer for the fist part. Any help for the first and second parts of the question ...
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1answer
21 views

Minimisation of Finite sum of a decreasing sequence

If $a_{1}<a_{2}<a_{3}<...<a_{n}$, find the minimum value of $$\sum_{i=1}^{n} (x-a_i)^{2}$$ Then find the value of $$f(x)=\sum_{i=1}^{n} |x-a_i|$$ Hi all, what would the best way be ...
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1answer
22 views

$C^{-1} (1+|x|^{2})^{\frac{s}{2}} \leq (1+|x|)^{\frac{s}{2}} \leq C (1+|x|^{2})^{\frac{s}{2}}$?

Let $s\in \mathbb R,$ and define $f: \mathbb R^{n}\to [0, \infty)$ such that $f(x)= (1+|x|^{2})^{\frac{s}{2}}, (x\in \mathbb R^{n})$ and $g:\mathbb R^{n}\to [0, \infty)$ such that $g(x)= ...
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31 views

Estimate $\displaystyle\left|\int_{\frac{\pi}{k}}^{\frac{\pi}{2}} (\sin \theta)^{-1+\frac{i(n+1)y}{2}} d\theta \right|$

I have to estimate the following integral $$\left|\int_{\frac{\pi}{k}}^{\frac{\pi}{2}} (\sin \theta)^{-1+\frac{i(n+1)y}{2}} d\theta \right|,\quad \forall k,n\geq 2 $$ According to Sogge (Oscillatory ...
2
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1answer
38 views

Del on Riemannian manifolds

I was supposed to figure out what $grad(div(e_r))$ is, where $e_r$ is the unit vector in spherical coordinates. In the following I assume $g:=diag(1,r^2,r^2sin^2(\theta))$ be the metric tensor. My ...
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26 views

Find an analytic function [duplicate]

Is it possible to find an analytic expression for a smooth, continuous, single-variable function $y=f(x)$ such that: 1) $f(0) = 0$ 2) $f(x_1) = y_1$ 3) $f(x) > 0, \forall x>0$ 4) ...
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1answer
23 views

differential equation as taylor series

Consider the equation $\frac{\mathrm{d} x(t)}{\mathrm{d}t} = g(x(t))$ , with $x(0) = x_0$, where g is function that admits derivatives of all orders.If the solution of the equation can be written as a ...
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4answers
76 views

A question about inequality ${(n+1)\over e^n}^n<n!$

How to prove the inequality $${(n+1)\over e^n}^n<n!$$ I have tried mathematical induction, but it doesn't work! Are there other methods to solve it?
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Prove that the convergence of the sequence (s3n) implies the convergence of (sn).

I write $s_n-s$, as $(s_n^3-s^3)/(s_n^2+s_n*s+s^2)$, true for all $n>N$. I'm trying to show that the denominator is convergent. But I don't know how to do this. Need help! Thanks. (Sorry about ...
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1answer
24 views

Prove that the convergence of the sequence (sn) implies the convergence of (s3n) [on hold]

Case 1: $s>0$. Assume $s>0$. Then there exists $N$ such that for all $n>N$ $s_n>0$. If $s_n^3-s^3$ converges to $0$, then write $s_n-s$, as was done in OH, as ...
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1answer
21 views

Does weak convergence of $\nu_{n}$ imply convergence of $\int{f_{n}(x)d\nu_{n}(x)}$?

Suppose that we know that $ \int{ |f_{n}(x) - f(x)| d\mu(x)} \longrightarrow 0 \qquad (1) $ for every probability measure $\mu \in \mathcal{A}$ in a certain class. Also, suppose that $\{\nu_{n}\}$ ...
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1answer
22 views

Is there a subset of R such that their Cantor-Bendixson rank is the first limit ordinal?

I'm looking for a set $A \subset \mathbb{R}$ such that $\bigcap^\infty_{n=0} A^{(n)} $ is a perfect set (i.e $X'=X$) but $\forall n \in \mathbb{N}$ the set $A^{(n)}$ isn't perfect (where ...
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15 views

To show closedness of a subset in a metric spaces

Let $(X, d)$ be a metric space and $p\in X$, $\delta>0$ be fixed. Let $$A=\{q \in X : p \in X, d(p,q)>\delta\}$$ How to show that $A$ is closed? I tried to show that directly by taking $A$'s ...
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2answers
29 views

Why is $\int_C {dz \over z - a} = 2 \pi i$ not a counter-example to Cauchy's theorem in a disk?

Cauchy's theorem in a disk states that if $\Delta$ is an open disk and $f$ is analytic on $\Delta$, then if $\gamma$ is a closed curve inside $\Delta$ we have that $$ \int_\gamma f(z)\ dz = 0 $$ ...
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1answer
20 views

Cauchy's theorem in a disk (Proof Verification)

Consider the following proof of Cauchy's theorem in a disk. My question is pasted at the bottom of the picture. (Note that in the proof below, a reference is made to "Theorem 2". In my textbook ...
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43 views

Computing $\int_\gamma { |dz| \over |z-a|^2}$

Goal: Compute $$ \int_{|z|= \rho} {|\mathrm{d}z| \over |z-a|^2} $$ under the condition $|a| \ne \rho$. Ahlfors' Hint: make use of the equations $z \bar{z} = \rho^2$ and $$ |\mathrm{d}z| = -i ...
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17 views

Set of limit points of Riemann Integrable functions

I've looked around for answers to this question. It seems like perhaps I don't have enough knowledge of functional analysis to figure out the answer (or even understand the answer), but I'm intrigued. ...
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51 views

Approaches to teaching and learning analysis

I found that studying linear algebra by getting into vector spaces and linear transformations first made things very easy. This is the approach Halmos or Axler, just to name a few, take. IMHO, the ...
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let E=C[X] be a normed space and T∈ L(E)… prove that.. [on hold]

Let E=C[X] be a normed space and T∈ L(E). And let $$\||P||_\ = \left\{ \sum ||P^{(n)}||_\infty, \; \; 0 \leq n \leq ∞ \right\}.$$ where $\||P||_\infty$=sup|p(x)|, 0≤x≤1 1- Justify that T:E→E ...
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With $\omega(\delta)$ being the modulus of continuity, prove $\omega( \delta_1 + \delta_2) \leq \omega(\delta_1) + \omega(\delta_2)$

If the modulus of continuity for the function $f: E \to \Bbb R$ is the function $\omega(\delta)$ defined for $\delta > 0$ by $$\omega(\delta) = \sup_{|x_1 - x_2| < \delta}_{x_1, x_2 \in \Bbb E} ...
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1answer
43 views

Show that if E is measurable set and f is continuous on E, then f(E) is measurable set

Please tell me how to prove or disprove it ! Show that if E is measurable set and f is continuous on E, then f(E) is measurable set
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27 views

show that f is integrable at $[a,c]$ and $[c,b]$

Let $f:[a,b] \to \mathbb{R}$ bounded and $c \in (a,b)$.Then $f$ is integrable at $[a,b]$ iff $f$ is integrable at $[a,c]$ and $[c,b]$.In this case,we have $\int_a^b f = \int_a^c f + \int_c^b f$. The ...
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38 views

Let $\,E = C([0, 1])$. Set $\,X =(E,||.||_\infty)$ and $\,Y =(E,||.||_1)$. … [on hold]

Let $E = C([0, 1])$. Set $\,X =(E,||.||_\infty)$ and $\,Y =(E,||.||_1)$. Let us consider the identity $I :X→Y$. Prove that I is continuous and bijective. Calculate $\,||I||$. Prove that $I^{-1}$ is ...
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28 views

Fundamental Theorem of Calculus and inverse..

If $F(x)$ is defined as $$F(x)= \int_{a}^{x} f(t) dt$$ calculate $(F^{-1})'(y)$ in terms of $f$. I have been working on this for a while now, does the aanswer to this incorporate the Inverse ...
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1answer
35 views

How to calculate the Laplace transform?

we know that ( http://en.wikipedia.org/wiki/Laplace_transform )\begin{equation} \mathcal{L} \{t^{5}f(t)\} = (-1)^5\frac{d^5}{ds^5} F(s)\end{equation} My question is \begin{equation} \mathcal{L} ...
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1answer
15 views

Nonexpansive Affine Operators in Hilbert spaces

Let $H$ be a Hilbert space with real inner product $\langle \cdot, \cdot \rangle : H \times H \rightarrow \mathbb{R}$. Let $T$ be an affine operator. Show that $T$ is nonexpansive, i.e., $\left\| ...
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1answer
36 views

Weierstrass Caratheodory on open interval

I have been working on this question for a while now, and if I have understood it correctly shouldn't the answer be that $\phi_{c}=f'(x)$ for all $x \in (a,b)$ as the function f , is now said to be ...
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1answer
28 views

How to interchange sum and integral when measure is in terms of Dirac measure?

Let $\{c_{k}\}_{k\in \mathbb Z} \subset \mathbb C$ such that, $\sum_{k\in \mathbb Z} |c_{k}| < \infty.$ Let $\delta_{k}$ is the unit Dirac mass at $k $, we note that $\mu = \sum_{k\in \mathbb Z} ...
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1answer
39 views

What is the limit of this function as $(x,y)$ approaches $(0,0)$?

Let the function $f \colon (\mathbf{R}^2 \setminus \{(x,y) \in \mathbf{R}^2 \colon x+y = 0 \}) \to \mathbf{R}$ be defined as follows: $$ f(x,y) \colon= \frac{xy}{x+y}$$ if $(x,y) \in \mathbf{R}^2$ ...
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44 views

Computing $\int_{|z|=2} {dz \over z^2 + 1}$

Goal: To compute $$ \int_{|z|=2} {dz \over z^2 + 1} $$ by decomposition of the integrand in partial fractions. Attempt: Let $\gamma$ be the circle around the origin of radius $2$. Let us ...
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2answers
22 views

How to quantify this statement

How do I quantify this statement? Let $x \in \mathbb{R}$. $1 \le x \le 2$ if and only if $1 \le x \le 1+ 1/n$ for some $n \in \mathbb{N}$. When I am trying to prove this, I am led (in the course of ...
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1answer
30 views

Showing the winding number of the unit circle is $1$

Let $\gamma$ denote the unit circle parameterized on the domain $[0,2\pi]$. I'm trying to compute $n(\gamma, 0)$ as follows: $$ n(\gamma,0) = {1 \over 2\pi i}\int_\gamma {dz \over z} = {1 \over 2 ...
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1answer
32 views

Invertible iff Bounded below and dense range

Statement: Given a Hilbert space $\mathscr{H}$ and $\mathscr{K}$ and a bounded operator $A \in \mathscr{B}(\mathscr{H}, \mathscr{K})$. Show that $A$ is invertible if and only if $A$ is bounded below ...
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75 views

Problem 6 - IMO 1985

For every real number $x_1$ construct the sequence $x_1,x_2,x_3,\ldots$ by setting $x_{n+1}=x_n(x_n+\frac{1}{n})$ for each $n \ge 1$. Prove that there exists exactly one value of $x_1$ for which $0 ...
1
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1answer
24 views

Definition of upper hemicontinuity of a correspondence.

When using and examining Kakutani's fixed-point theorem, I've got a question about upper hemicontinuity. A correspondence $f:X\rightarrow2^Y$ is a point-to-set mapping. One way to define upper ...
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3answers
20 views

Negation of continuity applied to a sequence

Show that if it is not true that $\lim_{x \to a} f(x)=l$ then $\exists$ $\epsilon$>0 and a sequence $(x_{n}) \rightarrow a$ as $n \rightarrow \infty$ such that $|f(x_{n})-l| \geq \epsilon$. Now ...
4
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2answers
29 views

Is there a word for describing “smoothness” quantitatively?

I've long wondered how to "quantitatively" describe how smooth a function is. For instance, a 1000 term Fourier series for a the Heaviside step function is technically smooth, as it has infinite ...