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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2answers
35 views

Where $f:[0,1]\to\mathbb R$ is Lebesgue integrable, show $\lim_{n\to\infty} n\lambda(\{x:|f(x)|\geq n\})=0$.

Title says it all. It's clear why $\lim_{n\to\infty}\lambda(\{x:|f(x)|\geq n\})=0$ -- since otherwise for arbitrarily high $n$ there'd be a subset of $[0,1]$ with nonzero measure where $|f|\geq n$ and ...
-1
votes
1answer
33 views

Analysis, limit of a complex sequence

Find the limit of $$z_n = n^2 \exp \left( \frac {\sin n + \textrm i \pi n^2} {4n \sqrt {n^2 + \textrm i + 1} } \right) \frac 1 {\sqrt {n^4 + n^2 + 2 \textrm i} }$$ and express your answer in the form ...
0
votes
1answer
29 views

Showing that the “abstract” tangent space of a submanifold of the $\mathbb{R}^d$ is isomorphic to the tangent space that's a subset of $\mathbb{R}^n$

Let $M$ be an $n$-dimensional smooth submanifold of the $\mathbb{R}^d$, and $p \in M$. Let $T_p^{A}M$ denote the "abstract" tangent space of $M$ in a point $p$, given by $T_p^AM = \{\gamma: ...
0
votes
1answer
31 views

Continuity of function varying rationals and irrationals

Let $f$ be defined as follows: If $x$ is irrational then $f(x)=0$ If $x$ is rational and expressed in its smallest form as $x=p/q$ then $f(x)=1/q$ I'm asked to prove that for irrational ...
1
vote
0answers
9 views

Metric of the doubling i.e circular billard table

Let $D^2=B_1(0) \cup_{\partial B_1(0)} B_1(0)$ denote the doubling in $\mathbb{R}^2$ i.e. the metric space which one gets after gluing two closed balls of radius $1$ with their induced metric coming ...
0
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0answers
20 views

Evaluation of a series with absolute value

I want to estimate or evaluate the series $$S(\xi)=\sum_{n=1}^\infty\beta_n\left|\sin(\pi n \xi)\right|,~~ \xi\in(l_0,l_1)$$ with $\beta_n=\frac{\omega\sin\left(\pi^2 n^2 ...
2
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3answers
66 views

Can $\pi$ or $e$ be a root of a polynomial with algebraic coefficients?

Since $\pi$ and $e$ are transcendental, neither can be the root of a polynomial with rational coefficients. However, it is easy to construct a polynomial transcendental coefficients (with $\pi$ or $e$ ...
1
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0answers
12 views

Extremum and nonsymmetric Hessian matrix

If $F$ is assumed only to have all second partial derivatives (but we don;t assume that they are continuos) than it could happen that the Hessian matrix in some stacionary point is nonsymmetric. Is is ...
0
votes
1answer
19 views

Total ordered set

I am not sure whether the relation $\leq$ for set $M=\{{f:\mathbb{R} \rightarrow \mathbb{R}}$: f Function} is a total ordered set, for $\leq$ is defined as: $ f \leq g : \leftrightarrow \forall x \in ...
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0answers
45 views

The Maximum Modulus Principle Applied to the Proof of Schwarz Lemma

I am using the following statement of the Maximum Modulus Principle: Theorem: Let $G$ be a region and let $f$ be holomorphic on $G$. Suppose $\exists~ a \in G$ such that $|f(z)| \leq |f(a)| ~ ...
2
votes
1answer
84 views

Conditioning a conditional probability to a sigma algebra

Suppose I have two random variables, $X$ and $Y$, defined on the space $(\Omega,\mathcal{F},P_1)$ which can both take the values $0,1,\ldots,N$. Suppose further I want to define the probability of ...
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0answers
7 views

Bounded input-Bounded output stability for countable system of ODES.

Let $X$ be a countably infinite dimensional Hilbert space. Let $f\colon X\to X$ be a compact, linear, symmetric positive definite map. Define an ODE as $u_t = -f(u-y)$ and $u(0) = 0$, where ...
0
votes
2answers
57 views

Show this harmonic function is constant

I'm trying to prove the following let $\alpha \in (0,1)$. If $u \in C^2(\mathbb{R}^n)$ is harmonic and $|u(x)| \leq \|x\|^{\alpha}$, Prove the $u$ is constant. Attempt to prove. Let's observe ...
0
votes
1answer
30 views

Prove that the derivatives with respect to different independent variables are also independent of each other

I don't know if this is a stupid question or not, but how one proves that given two independent variables (x,y), for which: $$ \frac{\partial x}{\partial y} = 0 $$ For any given function $f(x,y)$ it ...
1
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0answers
25 views

With $\lambda^*$ as the Lebesgue outer measure, $\epsilon\in(0,1),\ \lambda^*(E)>0$, find interval $I$ s.t. $\lambda^*(E\cap I)>\epsilon\lambda^*(I)$

We're to show that some interval $I$ satisfies the condition in the title. I.e., there exists an interval $I$ such that $\lambda^*(E\cap I)>\epsilon\lambda^*(I)$. I know that because any interval ...
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0answers
13 views

Is the norm a proper convex function?

Norms are closed convex functions. Are they also proper convex functions? Let $\Vert \cdot \Vert$ be a norm. To prove it is proper, it is sufficient to say that, since $$ \Vert 0 \Vert = 0$$, then ...
0
votes
1answer
14 views

Continuous functions and bounded sequence

Let $f:\Bbb R \to \Bbb R$ be a continuous function. Is the sequence $\{x_n\}\subset \Bbb R$ bounded, if $x=\lim f(x_n)$?
0
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1answer
23 views

A smooth function $f$, defined on an open ball in $\mathbb{R}^n$, can be written the sum of $n$ smooth functions with a certain property

Let $f: B \to \mathbb{R}$ be a $C^\infty$ function on an open ball $B := B_r(a) \subseteq \mathbb{R}^n$. I want to show that there exist $C^\infty$ functions $g_1, ..., g_n: B \to \mathbb{R}$ with ...
1
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0answers
25 views

Find norm of $T:(\ell^1,||\cdot||_1)\to(\mathcal C[0,1],||\cdot||_\infty),$ $(T(\xi))(x)=\sum_{k=0}^\infty a_k\xi_k x^k,$ $\xi\in\ell^1$

Let $a=(a_0,a_1,\cdots)$ be a fixed element of $\ell^\infty$. Define $$T:(\ell^1,\lVert\cdot\rVert_1)\to(\mathcal C[0,1],\lVert\cdot\rVert_\infty),\ (T(\xi))(x)=\sum_{k=0}^\infty a_k\xi_k x^k,\ \ ...
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0answers
22 views

If $F$ is a compact subset of a metric space $X$, prove $F$ is closed

I was asked to prove this in an exam. I began by proving that $X \setminus F$ is open. I assumed there was an $x\in X\setminus F$ such that $x_k\rightarrow x$. Then draw a ball of radius ...
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0answers
31 views

Differential of a complex 1-form

for a complex number $z$ consider the following 1-form $\omega=\frac{dz}{z}$ on $\mathbb{C}^*$. One can show that it is closed by explicitely calculating the differentials of the real and imaginary ...
0
votes
1answer
47 views

$\varphi:B\times A\rightarrow\mathbb{R} $ defined by $\varphi (f,x)=f(x) $is continuous.

Problem says: Let $A\subset\mathbb{R}^{n}$ be compact and let $B\subset\mathcal{C}(A,\mathbb{R})$ be compact. Show that there are an $f_{0}\in B$ and an $x_{0}\in A$ such that $g(x)\leq ...
1
vote
1answer
24 views

All Derivatives Bounded from Below

Is it possible to construct a function $f:\mathbb{R} \to \mathbb{R}$ such that there is $c>0$ with the property that for each $n$ and each $x \in \mathbb{R}$ we have $f^{(n)}(x) \geq c$? If not, ...
0
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0answers
20 views

Relation between the eigenvalues of $\Delta$ and counting lattice points

I was reading a paper with the following information: "Let $\mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n$ be the flat torus, let $\varphi$ be the eigenfunctions and $\lambda$ the eigenvalues of the ...
0
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0answers
43 views

Compute the radius of convergence and interval of convergence of $\sum_{n=1}^\infty\left (\frac {4+2(-1)^n} 5\right)^nx^n $

Compute the radius and interval of convergence of: $$\sum_{n=1}^\infty \left(\frac {4+2(-1)^n} 5\right)^nx^n .$$ I went about this question by applying the root test and this is what I have gotten ...
0
votes
1answer
26 views

Ultrafilters voting system counter example

I know that we can view ultrafilters as a counter to Arrows' theorem. See for example https://terrytao.wordpress.com/2007/06/25/ultrafilters-nonstandard-analysis-and-epsilon-management/ or ...
0
votes
2answers
31 views

counterexample for $C^1(U)$ not complete in any dimension

Cleary $C^1[a,b]$ is not complete with $\|\cdot\|_{\sup}$. I am looking for a counterexample which is working in any dimension, i.e. $C^1(U)$ is not complete for any open $U\subseteq \mathbb R^n$ ...
4
votes
0answers
86 views

A conjectured asymptotic expansion of a function related to the sine and cosine integrals

Recall the definitions of the sine and cosine integrals:$$\operatorname{Si}(x)=\int_0^x\frac{\sin t}t dt,\quad\operatorname{si}(x)=-\int_x^\infty\frac{\sin t}t ...
2
votes
0answers
17 views

hyperreal numbers limited number multiplied by infinitesimal

I'm trying to prove this basic result about the hyperreals: If x is infinitely close to y i.e. x-y is an infinitesimal and b is a limited number i.e. r < b < s where r and s are real numbers, ...
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votes
0answers
24 views

Numerical Analysis I: problem with interpolation

We want to build a table of values of the function $f(x)=\dfrac{x^4-x}{12}$, such that the linear interpolation for $0\leq x\leq a$ has an error inferior to $\delta$. Determine which will be the ...
1
vote
1answer
26 views

Relation between Compactness, Closedness and Completness of metric spaces

I would like to know as many relations as possible to get a better picture. I know that if $f$ is continuous and $(X,d)$ is complete, then $f(X)$ is complete $\iff$ closed. Question:However, are ...
2
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0answers
14 views

Finding lower $\ell_{q}$ estimates for weighted scalar subsequences

Here is a stumper. Define $c_{00}$ as the space of all finitely-supported real sequences, i.e. all sequences $(a_n)_{n=1}^\infty\in\mathbb{R}^\mathbb{N}$ such that $a_n\neq 0$ for only finitely many ...
0
votes
0answers
38 views

$ \frac{\ln x\ln \alpha\beta x}{\ln^{2} \alpha x}\leq\frac{\alpha(x-1)(\alpha\beta x -1)}{\beta(\alpha x -1)^{2}} $

The following inequality is obviously true for $\frac{\alpha}{\beta}$ greater than 1,but i do not know for which value of the ratio $\frac{\alpha}{\beta}$ less than 1 , $$ \frac{\ln x\ln \alpha\beta ...
1
vote
1answer
14 views

Drawing $D=\{(x,y)| |x-1|+|y-1|\leq \frac{1}{2}\}$ in $\mathbb R^2$

$$D=\{(x,y)| |x-1|+|y-1|\leq \frac{1}{2}\}$$ I know how $y=|x-1|$ looks, and $|y-1|$ but how do I find this? The problem I have is to find $$\iint_{D}|\ln(xy)|dxdy$$. I think I can do this, just need ...
11
votes
3answers
1k views

Is it always safe to assume that a integral is zero if it has equal bounds?

I'm still a "newbie" on mathematical analysis and I stumbled upon this integral. This is my solution: $$\int_0^{2\pi}{\frac{x\cos x}{1+\sin^2x}dx}$$ Now I substitued with $t=\sin x$ ...
2
votes
2answers
36 views

injective open map between two euclidean spaces

Does there exists an injective function from $R^2\ to\ $R such that image of an open set is open ? Where $R^2$ and $R$ are usual euclidean spaces. Please help.Thank you.
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0answers
34 views

Theorem 3.22 in Baby Rudin: Is this proof correct?

Here's Theorem 3.22 in the book Principles of Mathematical Analysis by Walter Rudin, third edition. $\sum a_n$ converges if and only if for every $\varepsilon > 0$ there is an integer $N$ such ...
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0answers
22 views

What does the explicit formula means in this sentence?

I'm reading Classic and Multilinear Harmonic analysis vol.2 - Muscalu, Schlag In page 134, it says, $$\int_{\partial B(x,\epsilon)}-F(y-x)\frac{\partial u}{\partial\nu}(y)d\sigma(y)=0$$ can be ...
1
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1answer
59 views

Is this proof correct? (Showing that if $f$ is zero except on a closed set $E$ of null measure then $f$ is integrable

Let $E \subset I\times I$, where $I = [0,1]$, and suppose that $E$ has null measure and is closed. Then I want to prove that a bounded function $f : I\times I \to \mathbb{R}$ that is null except on ...
1
vote
2answers
44 views

How can I show that this function is riemann integrable?

Let $I = [0,1],$ and $f : I\times I \to \mathbb{R}$ defined as follows: $$f(x,y) = \frac{1}{q} ~~\text{if}~~ x = \frac{p}{q}, y \in \mathbb{Q},$$ $$f(x,y) = 0 ~~\text{otherwise}$$ How can I show ...
1
vote
0answers
18 views

Is this question well-formed? Let $T\in\mathcal B(X,Y),\ \lVert T\rVert<1,\ Y$ Banach, show $\sum_{n=0}^\infty T^n\in\mathcal B(X,Y)$.

If $X=Y$, then I think I have solved this problem entirely already. But if $X\neq Y$, then I don't understand what is being asked. How is $T^n$ defined when $X\neq Y$? When I consider $X=Y$, then I ...
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0answers
55 views

Need help proving that $\sqrt{x^2}\leq \sqrt{x^2+y^2}$

So, I'm working through Rudin's Principles of Mathematical Analysis, and I'm stuck on the proof of theorem 3.4, stated as follows. I'm working through the $(\Rightarrow)$ direction, that is, ...
0
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0answers
19 views

$\sqrt[n]{x}$ as a power series in a complete field with absolute value.

Let K be a field complete with respect to a non-archimedean absolute value $| \cdot|:K^*\rightarrow \mathbb{R}$ and $char(K)=0$ or $char(K)\nmid n$. I want to prove that if $|x|\leq 1$ (i,e $x$ is in ...
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votes
3answers
39 views

True or false: problems on sequences

True or false (if true justify): if $\{x_n\}$ and $\{x_ny_n\}$ are bounded sequences then $\{y_n\}$ is bounded. if $\{x_n\}$ and $\{y_n\}$ are sequences such that $x_ny_n \rightarrow 0$ ...
0
votes
0answers
11 views

Modulus of continuity example

could help me clear something out. I am looking for an example of modulus of continiuity. I was told that is doable for: $f(x)=\sqrt{1-x^2}$ for $x\in [0,1]$. Some calc: From ...
1
vote
0answers
24 views

Prove that for a homogeneous function of degree one all directional derivatives exist

I am trying to prove that for a function $f:\mathbb{R}^n \rightarrow\mathbb{R}$ that is homogeneous of degree one, all directional derivatives exist. I also want to prove that it is differentiable if ...
0
votes
0answers
16 views

Example of a function that is zero out a null measure set that is not continuous out of this set

Let $f : Q \subset \mathbb{R}^n \to \mathbb{R}$ a function that is $0$ outside of a set $ E \subset Q$ with measure of $E$ equals $0$. I mean, $E$ is a null measure set. ($Q$ a closed rectangle). ...
1
vote
4answers
51 views

Banach Contraction mapping of $\Phi(f)(x)=\int_0^x \frac{1}{1+f(t)^2}dt$ , find a fixed point.

Let $(X,d)$ be metric space with $d(f,g)=\sup |f(x)-g(x)|$ where $X$ is the set of continuous function on $[0,1/2]$. Show $\Phi:X\rightarrow X$ $$\Phi(f)(x)=\int_0^x \frac{1}{1+f(t)^2}dt$$ has a ...
0
votes
0answers
11 views

How can I prove $ S'-S $ is a set of boundary points of $S$?

How can I prove $ S'-S $ is a set of boundary points of $S$ where $S'$ is the set of all limit points of $S$? I am proving $S^0 \cup bd(S) \supseteq S \cup S'$ by contradiction. I supposed to the ...
1
vote
1answer
36 views

How can I prove that $S=\{x_k:k\in \Bbb N\}\cup\{x_0\}$ is closed in $\Bbb R^n$?

How can I prove that if $\{x_k\}$ is a convergent sequence in $\Bbb R^n$ with limit $x_0$, then $S=\{x_k:k\in \Bbb N\}\cup\{x_0\}$ is closed in $\Bbb R^n$? This is my attempt: It suffices to show ...