# Tagged Questions

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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### 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 ...
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### 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 ...
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### 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 $e$ ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 $x=\lim f(x_n)$?
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
### 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 ...