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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12 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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14 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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5 views

Typo in Caffarelli-Silvestre?

I am reading two papers by Caffarelli and Silvestre, namely Regularity results for fully nonlinear equations by approximation and The Evans-Krylov theorem for nonlocal fully nonlinear equations. From ...
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1answer
15 views

Select a subsequence to obtain a convergent series.

Does there exists a subsequence of $\mathbb N$ which is denoted by $a_k$,s.t. $$\sum_{k=1}^{\infty}\frac{1}{(\log a_k)^{1+\delta}}\lt \infty$$ $\delta$ is a small postive number($\ll 1$) and ...
3
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2answers
34 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 $\sum a_k b_k$ $<$ $S\sum a_k$. ...
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33 views

Visual understanding of this convergence

In these lecture notes by Ueltschi here, I found in Definition 2.3 a peculiar type of convergence. Especially the second property is hard for me to visualize what it means, could anybody try to ...
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1answer
22 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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26 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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1answer
17 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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0answers
31 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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1answer
10 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 ...
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2answers
23 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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0answers
50 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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1answer
50 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
21 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 ...
2
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1answer
23 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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0answers
8 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 ...
2
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1answer
33 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 ...
2
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0answers
25 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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37 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 ...
2
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1answer
29 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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0answers
18 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
51 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 ...
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2answers
69 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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0answers
15 views

Mean value theorem and Multi-valued functions

Let a point $x$ map to a set of points $\{y | y \in U \}$ where $U \subseteq \mathbb{R}$ or $U \subseteq \mathbb{C}$. Can MVT be generalized to multi-valued functions ?
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0answers
15 views

Generator of smooth complex-valued functions vanishing at infinity

Let $C_0(\mathbb{R})$ be the $C^{\ast}$-algebra of complex-valued functions vanishing at infinity, with involution given by $f^{\ast}(x) = \overline{f(x)}$. How can I prove that this commutative ...
2
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1answer
38 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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41 views
+50

Exemple about the difference between Morse and degree theory

i found this example but i don't understand how we applyed Morse theory and why we can't applyed degree theory. if the functional $f$ behaves like $<lu,u>$ at infinity where the symmetric ...
2
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2answers
71 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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0answers
10 views

Extension of norm from integral domain to fraction field

Let's say I have an integral domain $D$ with norm $\|\cdot\|:D\to\mathbb{R}.$ I aim to show that $\|\cdot\|$ can be uniquely extended to $\text{Frac}(D).$ I think the simple extension ...
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1answer
19 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
27 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
50 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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45 views

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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0answers
52 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} + ...
2
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1answer
24 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} ...
0
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0answers
26 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 ...
0
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1answer
19 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}} ...
1
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1answer
15 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 ...
0
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1answer
23 views

rate of change and the “1 - 10” scale [on hold]

I am using the 1 - 10 scale for pain management in therapy. And I was wondering, when a patient says there pain is a 4 out of 10 one week, then the next week the say it's down to a 1 out of 10, that ...
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0answers
10 views

Show that I is a non-empty interval if every continuous function has an interval as its image

Assuming $I$ is a non-empty interval of real numbers I want to show that for any continuous function $f$ that $f(I)$ is also an interval. So given, say, $y_{1}, y_{2}$ in $f(E)$ and without loss of ...
2
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1answer
44 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!
0
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1answer
29 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
54 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 ...
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0answers
29 views

Finding the source for minimizer of a functional for all $C^2$-curve $x(t_0)=x_0$ and $x(t_1)=x_1$

I am trying to find where this problem comes from and its corresponding proof for my students, but I cannot find the source anywhere. If anyone can find the source of this, or has any ideas where I ...
2
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1answer
18 views

Question about Hilbert transform(applying plancherel theorem)

Let $f\in S(\mathbb{R})$(Schwartz function on real line). Then Hilbert transform $H$ of $f$ is defined by $\displaystyle Hf(x)=\lim\limits_{t\rightarrow0}\int_{|y|>t}\frac{1}{y}f(x-y)\,dy$ One ...
1
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1answer
18 views

Vector Analysis (Parametized curve)

The question is find a familiar parameterized curve that has the property $r(t) \times\dfrac{dr}{dt}=0$. The only curve that I can see that works is the line through the origin. I was just wondering ...
0
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0answers
32 views

Compactness criterion

I have this compactness criterion and I want to apply it, but I don't know what I must write to see if (a) is satisfied and also for (c)? For a subset $H\subset\mathcal{BC}(\mathbb{R},Y)$ to be ...
0
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0answers
17 views

Finding Minimum Distance of a Point from Curve

While finding the distance of a point from a curve (which is graph of a function), the usual method I saw is as follows: given a point and a curve $\{x,f(x)\}\colon c\in\mathbb{R}$ (where ...
0
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4answers
61 views

Proof that the set of irrational numbers is dense in reals

I'm being asked to prove that the set of irrational number is dense in the real numbers. While I do understand the general idea of the proof: Given an interval (x,y) choose a positive rational ...