Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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Subsets of a metric space in which Hausdorff semi-distance is symmetric

These are the definition of Hausdorff distance and Hausdorff semi-distance for subsets of a metric space $X$. ‎‎Hausdorff semi-distance of two subsets ‎$‎A‎, B‎ \subset X$ is defined as below: ‎$‎d(A ...
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50 views

bounding supremum with Sobolev embedding theorem

I consider $d \geq 2$ and the domain $ D = [0,1]^d$. I consider a function $f : D \to \mathbb{R}$ that is $k$ times continuously differentiable. [$k$ can be specified as large as possible.] I am ...
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3answers
192 views

Infimum of a set with two variables

I have encountered a problem concerning the infimum of a set: Prove that. $$\mathrm {inf} \left\lbrace\sqrt{a^2+{1\over b^2}}:a,b\in(0,1) \right\rbrace=1$$ What I've been able to do is to prove that:...
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273 views

How prove this $\int_{a}^{b}f(x)dx=\frac{1}{2}(b-a)[f(a)+f(b)]-\frac{1}{12}(b-a)^3f''(\xi)$

Let $f(x)$ be a twice-differentiable function on $(a,b)$,show that there exsit $\xi\in(a,b)$ ,such $$\int_{a}^{b}f(x)dx=\dfrac{1}{2}(b-a)[f(a)+f(b)]-\dfrac{1}{12}(b-a)^3f''(\xi)$$ if this problem ...
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72 views

What is a everyday example of a non measurable set? [duplicate]

I'm working on my understanding of measurable sets and my immediate intuition wants to know what's not a measurable set? Initially I think of some space where divisions go to infinity, like a ...
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265 views

Prove that Archimedean Property implies that $lim_{n->\infty} 1/n$ =0

I am very curious how to prove this. To start off, we assume the Archimedean Property, or there exists an $\epsilon>0$ s.t. for a natural number n, 1/n < $\epsilon$. But From there I am simply ...
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104 views

$f(x)$ is everywhere differentiable on $[a,b]$ then give examples

$f(x)$ is everywhere differentiable on $[a,b]$ then give examples for each (they are independent) (1) $f'(x)$ is not Riemann integrable (2) $f''(x)$ does not exist (3) $f'(x)$ is not continuous
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144 views

Writing the roots of a polynomial with varying coefficients as continuous functions?

Consider the monic polynomial $$p_{\zeta}(z) = z^n + a_{n-1}(\zeta)z^{n-1} + \dots + a_0(\zeta), $$ where the $a_{i}$'s are continuous functions defined over $\mathbb{C}$. As is well known, the ...
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28 views

What assumptions should be made?

take a problem like A trough is 12 feet long and 3 feet across. Its ends are isosceles triangles with altitudes of 3 feet. Water is being pumped into the trough at 2 cubic feet per minute. How fast ...
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303 views

Picard Iterates Converge Uniformly

I have a homework question that asks to show that the Picard iterates $$ \phi_{n+1}(t) = \int_0^t 1 + \phi_n^2(s) \, ds, \quad \quad \phi_0(t) = 0 $$ converge uniformly on any compact interval $[-r, r]...
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52 views

Finding the set of analytic functions whose image is a subset of a given set

Let $A=${$z\in\mathbb{C}||z|=1$} and $B=${$z\in\mathbb{C}||z|<2$}. I want to find the the set of analytic functions such that $f(B)\subset A$. Is there a way to solve this? Hope someone could help ...
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2answers
58 views

Select a subsequence to obtain a convergent series.

Does there exists strictly increasing sequence $\{a_k\}_{k\in\mathbb N}\subset\mathbb N$, such that $$ \sum_{k=1}^{\infty}\frac{1}{(\log a_k)^{1+\delta}}\lt \infty, $$ where $\delta>0$ given and $$...
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85 views

Convergence of $\sum{a_kb_k}$ if $\sum{a_k}$ converges and $\sum{b_k}$ absolutely converges.

Convergence of $\sum{a_kb_k}$ if $\sum{a_k}$ converges and $\sum{b_k}$ absolutely converges. I tried to think that Since $\sum |b_k|$ is bounded I thought that $\sum a_k b_k$ $<$ $S\sum a_k$. Is ...
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1answer
70 views

Question about the answer to Kac's problem: 'Can one hear the shape of a drum?'.

I'm looking at the article of Gordon, Webb and Wolpert http://www.math.upenn.edu/~kazdan/425S11/Drum-Gordon-Webb.pdf, having only basic notions of group theory. In this article the authors describe ...
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62 views

The differentiability of the complex valued function $(Rez)(Imz)z\over|z|^2$

$$ f(z) = \left\{ \begin{array}{ll} \Re(z)\Im(z)z\over|z|^2 & \quad z \neq 0 \\ 0 & \quad z = 0 \end{array} \right. $$ I want to prove that this ...
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37 views

Square integrability

Given a function $g(y)=\int_y^{\infty}f(x) dx$ and given that I know that for $y\rightarrow-\infty$ the function $g(y)\rightarrow C$, where C is a constant, why is the last condition implying that the ...
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50 views

Algebra of limits- Is this proof correct?

If you go to http://math.wikia.com/wiki/Algebra_of_Limits Shouldn't the line before the last line read $$\lim_{n \to \infty} \frac {1}{y_{n}} = \frac {1}{y}$$ Instead of $$\lim_{n \to \infty} \frac ...
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57 views

Inequality in Evans PDE section 5.7

I'm stuck in the proof of the Compactness Theorem in Evans PDE 2nd edition book. On page 287, last line, how do you get the inequality $$ \epsilon \int_{B\left(0,1\right)}\eta\left(y\right)\int^{1}_{0}...
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32 views

Is the following true for $f: \Bbb R^3 \rightarrow \Bbb R$ continuous?

For $f: \Bbb R^3 \rightarrow \Bbb R$ continuous I am asked to prove that if there is an $x$ such that $f(x)=0$ but $f(o,o,o)$ is not zero. Then there exists another $y$ closer to the origin, such ...
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62 views

Conditions for convergence of $\sum\limits_{n=1}^\infty{a^nf(n)}$

assume $a>0$, and for all $n$ we have $0 \leq f(n) \leq 1$. Is there a necessary and sufficient condition on the series $f(n)$ for which $\sum\limits_{n=1}^\infty{a^nf(n)}<\infty$ ? Thanks!
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100 views

Prove that $\{\sqrt[n]{e^{n+1}}\}$ is convergent.

I need to prove that $\{\sqrt[n]{e^{n+1}}\}$ is convergent, and find its limit using this theorem: Let $f:E \to R$ with $x_{0} \in E$ an accumulation point of $E$. Then these are equivalent: 1)$f$ ...
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79 views

Correctness of proof that $\lim_{n\to \infty}\sqrt n*c^n=0$

My proof is as follows: Assume $|c|\lt 1$ and $c$ can be written as 1/1+d for d>0 The definition of the mentioned limit is: For all $\epsilon>0$ there exists a natural number N s.t. for all n $\...
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1answer
497 views

Proof that the set of irrational numbers is dense in the reals

The hint I was given was to simply prove that y=xz is irrational given that x is nonzero, x is rational and z is irrational. Here's how I did it: Claim: y=xz is irrational Proof: Assume $x\neq0$, x ...
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1answer
52 views

Correctness of proof that an ordered field S that has the supremum property also has the infimum property

First question I have is how would you describe the relationship between an ordered field and an ordered set and continue the proof by treating the field as a set? I want to say that right in the ...
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20 views

Why are weak-mixing systems considered “random” and compact systems considered “ordered”?

As I understand it, weak-mixing systems sort of tend to become "orthogonal" to themselves on the long run, and compact systems tend to become almost periodic. How is this related to them being called "...
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48 views

Does $\lim_{x \to 0}({z^2\over \overline z})$ exist? $(z\in \mathbb{C})$

I am trying to figure out if $\lim_{x \to 0}({z^2\over \overline z})$ exists or not. This is a way I though to show that this does not exist but I am not entirely sure. Let $a_n={1\over n}$ and $b_n=...
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103 views

Quotient of a Banach space $X$ gets quotient topology under standard norm induced from $X$.

Let $X$ be a Banach space and $Y$ be a closed subspace of $X$. Then there is a canonical norm on $X/Y$. I want to show that this norm induces quotient topology on $X/Y$. Any hint/solution? I was ...
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38 views

In which metric spaces other than the discrete spaces are the closures of open balls different from closed balls?

Let $(X,d)$ be a metric space such that $d$ is not the discrete metric. Let $x_0 \in X$, let $r>0$, and let $$B(x_0;r) \colon= \{ x \in X \colon d(x,x_0) < r \}$$ be the open ball with center $...
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71 views

Norm equivalence of p-forms

Let $U$ be a bounded domain in the Euclid space $(\mathbb{R}^d,g)$. $g_{\wedge^p}$ denotes the fiber metric of $\wedge^pT^{\ast}\mathbb{R}^d$ derived from $g$. $A^p$ denotes the set of p- forms on ...
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1answer
1k views

Distance between a point and a closed set in metric space

Here is what I am thinking. Let (X,d) be a metric space and let C be a closed subset of X. Fix any poin p in X. Then, there exists a point q in C such that d(p,q) = distance(p,C). I think this ...
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32 views

How to compute the derivative of this functional on a manifold?

I'm a little puzzled by the following computation. Suppose $\Sigma$ is a compact two-dimensional Riemannian manifold and $M$ is any Riemannian manifold. Let $\Omega$ be some space of functions from $\...
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1answer
358 views

Continuous iff Oscillation is zero

For a bounded function $f:D\subset \Bbb R^n \rightarrow \Bbb R$, $b$ in $\Bbb R^n$, and a real number $\delta>0$. Define the following: $M(f,b,\delta)$=sup{f(x)$: x$ in $D$ and $||x-b||<\...
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188 views

Prove that $X$ is complete but not inner product, and vice versa

Let $X$ be the space $C[0, 1]$ under the norm $||·||_{p}$ for $1 \leq p \leq \infty$. (a) Show that $X$ is complete for $p = \infty$, but it is then not an inner product space. (b) Show that $X$ is ...
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1answer
45 views

Prove that for $0<p<1$, $|x-y|^p$ is a metric space on $\mathbb R^{n}$

Define the function $f_p : \mathbb R^{n} \to \mathbb R^{n}$ for $n ≥ 2$ by $f_p(x) = \sum_{k=1}^{n} \lvert x\rvert^{p}$. Show that for $0<p<1$, we get $d_b(x,y) = f_p(x-y)$ is a metric on $\...
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1answer
88 views

Generator of complex-valued functions vanishing at infinity

Let $C_0(\mathbb{R})$ be the $C^{\ast}$-algebra of continuous complex-valued functions vanishing at infinity, with involution given by $f^{\ast}(x) = \overline{f(x)}$. How can I prove that this ...
2
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1answer
47 views

Prove that any derivative of a given function is bounded

Let the function $f\left( x \right) = \left( {\frac{{1 - \cos x}} {{{x^2}}}} \right)\cos (3x)$ if $x\ne 0$ and $f(0)=\frac{1}{2}$. Prove that any derivative of $f$ is bounded on $\mathbb{R}$. Thank so ...
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53 views

How to prove this assertion about $\mathbb{R}^k$?

Suppose $k \geq 3$, $x$, $y \in \mathbb{R}^k$, $|x-y| = d > 0$, and $r > 0$. Then how to prove the following assertions? (a) If $2r > d$, then there are infinitely many $z \in \mathbb{R}^k$ ...
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107 views

Does the series $\sum_{n=1}^\infty (-1)^n \frac{\cos(\ln(n))}{n^{\epsilon}},\,\epsilon>0$ converge?

Numerical results showed that the series $$Q=\sum_{n=1}^{\infty} (-1)^n \frac{\cos(\ln(n))}{n^{\epsilon}},\epsilon>0\tag{1}$$ with $\epsilon=10^{-6}$ converged to $-0.53259554096828...$ We can ...
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1answer
26 views

How do induction on a recursive defined function?

I made a mistake on some homework, because I didn't prove by induction, but I am lost upon how to prove this my induction. How I understand in induction: Show base case (n=1) is true Assume n ...
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1answer
42 views

Why $f (x):= \frac{1}{\sqrt{x}\left(1+\left|\ln x\right|\right)}$ only belongs to $L^2(0, \infty)$

This is a result given in Royden and Fitzpatrick (p. 143). Show that $$ \int_0^\infty \left[ \frac{1}{\sqrt{x}\left(1+\left|\ln x\right|\right)} \right]^p < \infty $$ if and only if $p=2$. That ...
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1answer
87 views

Understanding the term “Abstraction” in mathematics

When the need for abstraction is asserted in mathematics is it generally meant that there is a need to apply a definition to n-dimensions such that n is an integer going to infinity?
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How to prove that $b^{x+y} = b^x b^y$ using this approach?

Fix $b>1$. If $m$, $n$, $p$, $q$ are integers, $n > 0$, $q > 0$, $r = m/n = p/q$, then I can prove that $(b^m)^{1/n} = (b^p)^{1/q}$. Hence it makes sense to define $b^r = (b^m)^{1/n}$. I ...
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70 views

How to find if an integral is possible to compute: Failing to solve integral for quadratic functional

I am trying to solve the below integral, and no computational method seems to be capable of solving this, nor can I do it by hand. Any ideas? $$\int_{t_0}^{t_1}[a(t)((2\dot{x^*}\dot{\eta} + \dot{\...
2
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1answer
57 views

Fourier Transforms of $L^1$ functions

Suppose that $f_n$ and $f$ are $L^1(\mathbb R^n)$ functions with $f_n \to f$ in $L^1$ sense. Then is it true that their Fourier transforms defined as $$ \hat f(\xi) := \int_{\mathbb R^n} e^{-ix\cdot\...
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75 views

Point-set topology - derived sets, boundaries, etc.

Consider the metric space $(\mathbb R^2,\sigma)$, where $\sigma$ is the discrete metric. Let $$E=\lbrace(x,y)\in \mathbb R^2 : x^2+y^2<1 \rbrace$$ i.e. the unit disc. Find the following sets [note ...
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1answer
40 views

Is the split normal distribution analytic on $\mathbb{C}$?

I wonder if the split normal distribution which expressed as following is analytic on $\mathbb{C}$ or not? $ p(x)= \left\{ \begin{array}{l l} \frac{2}{1+\gamma} \cdot \frac{1}{\sqrt{2 \pi}} \exp{...
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1answer
19 views

If $f$ is $C^1(U)$) , are $D_i f_j$ where $i=1,\ldots,n$ and $j=1,\ldots,m$ are all continuous on $U$?

$f$ is a function from an open set $U$ in $R^n$ to $R^m$ then $f=(f_1,f_2,\ldots,f_m)$, I am confused whether the following are true: If $f$ is continuous on $U$, does that imply that $f_1,\ldots,...
2
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1answer
85 views

Is every closed ball (or open ball) in the Eucledean Space $R^n$ convex?

I am solving a problem and I need to use this fact: Every closed ball (or open) in the Eucledean Space $R^n$ convex? Hoever, I am not sure if it is true or not. Can anyone help? Thanks!
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1answer
102 views

Bounding an integral in the proof of solution to Poisson's equation

I'm trying to understand the proof that $u(x) = \int_{\mathbb{R}^n} \psi(x-y)f(y)dy$ is indeed a solution to $-\Delta u = f$. I can follow it if I do it in a particular dimension ie 2. However, in ...
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2answers
92 views

continuity and limit of a function.

Below is the question: To what degree would the sequence definition of continuity need to be modified in order to be suitable as a definition for the limit of a function? In other words,if $f$...