# Tagged Questions

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### Correctness of the proof that the set $\{x \in \mathbb{Q} : x>0 \text{ and } x^2>2\}$ does not have a smallest element

Let $F=\{x \in \mathbb{Q} : x>0 \text{ and } x^2>2\}$. I am asked to show that $F$ does not have a smallest element. The hint is to simply prove the claim: 'If $p$ is a rational number in ...
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### convergence of a sequence of cauchy

I would like to ask something about the convergence of a Cauchy sequence in a space $X$ metric. There will be a metric space $X$ such that If $(x_n)$ cauchy sequence in $X$ then $(x_n)$ is not ...
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### How to show if $a$$\leq$$b_1$, for every $b_1>b$, then $a$$\leq$$b$ where a,b$\epsilon$R?

Not positive on the proper approach to this problem. My first thought: $a$ $\leq$ $b_1$ means either $a=b_1$ or $a<b_1$. Should it broken up into cases? Second attempt: Assume, to the ...
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### Question on a derivation regarding the non-linear ODE $x'' = -U'(x)$, $U$ potential

Let $U$ be a potential function, and consider the IVP $$(*) \quad x'' = -U'(x), \qquad x(t_0) = x_0, \quad x'(t_0) = v_0.$$ We suppose the following: (V) Let $x_0, v_0$ be initial values and let ...
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### Show that there exists an $\sum$-measurable simple function $\phi$ such that: $\int |f-\phi| d\mu <\epsilon$

Problem: Let $f \in L(X;\Sigma)$ where $L(X;\sum)$ is the set of integrable functions that can be written as $f=f^{+}-f^{-}$ where $\int f^{+} d\mu < \infty$ and $\int f^{-} d\mu < \infty$ ...
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### Showing that if the $n$th derivative of a function is bounded then it is real analytic

I reproduce from my lecture notes: Suppose $f$ is $C^\infty$ on $[a,b]$ with $$\left|f^{(n)}(x)\right|\leqslant M~~\text{for all}~~x\in(a,b).$$ Then $f$ is real analytic in $[a,b]$. Proof. ...
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### Derivative nonlinear Schrodinger equation

I'm dealing with the following DNLS $$iu_t+u_{xx}=i(|u|^2u)_x$$ Let's consider the following transformation $w=\exp(-i\int_{-\infty}^x|u|^2dy)u$. I'm interested in the equation satisfied by $w$. I ...
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### Lipschitz constants of projections

Consider two compact sets $A, B \subset \mathbb{R}^n$. Assume that the projection mappings $P_A: \mathbb{R}^n \rightarrow A$, $P_B : \mathbb{R}^n \rightarrow B$ have Lipschitz constant $1$ and $L$, ...
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### Green's function to operator

I would like to understand how one can show that the Green's function in this table is a Green's function to the D'Alembert operator? I refer to the wikipedia page about Green's function
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### Projection of a set $G_\delta$

The canonical projection $\pi:\mathbb{R}^2\rightarrow \mathbb{R}$ such that $\pi(x,y)=x$ maps $G_\delta$ sets to Borel sets? i.e. If $A=\cap_n^\infty A_n$ with $A_n$ open sets, then $\pi(A)$ is ...
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### Show that for any $g \in L_{p'}(E)$, where $p'$ is the conjugate of $p$, $\lim_{k \rightarrow \infty}\int_Ef_k(x)g(x)dx = \int_Ef(x)g(x)dx$

Let $1 < p < \infty, f_k \in L_p(E), k = 1, 2, ...,$ and $\lim_{k \rightarrow \infty}f_k(x) = f(x)$ a.e., $\sup_{1 \leq k<\infty}||f_k||_p \leq M$. Show that for any $g \in L_{p'}(E)$, ...
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### What can be said about an infinitely differentiable function whose Taylor series diverges?

What can be said in general of an infinitely smooth function whose Taylor series diverges? According to Borel Theorem it is possible to construct such but what kind of property have those special ...
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### Measure theory problem from Stein real analysis

Let $\mu$ be a Borel measure on the sphere $S^{d-1} = \{x \in \mathbb{R}^d:|x|=1\}$ which is rotation-invariant in the sense that $\mu(r(E)) = \mu(E),$ for every rotation $r$ of $\mathbb{R}^d$ and ...
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### About a convergence of measurable functions

Let $f_{n}$ be a sequence of measurable functions in M(X,m), is that true that {${x∈X∣lim f_{n}∈R}$}  = $⋃ _{M=1} ^∞⋂ _{N=1} ^∞ ⋃ _{n=N}^ ∞${x∈X∣ ∣f_{n} -f_{N} ∣< (1/M)} and that ...
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### Is this a B-measurable funtion?

Is the function defined by: $f(x)=e^x$ if x is in E and $f(x)=e^{-x}$ if x is not in E measurable?, here f goes from R to R, and E is not member of the Borel-algebra
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### Some special Metric on R

Apart from usual and discrete metric is there a metric on R which satisfy: d(x, y) = d(x+r , y+r) where x and y are any real no. and r is arbitary real no. Similarly is there a ...
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### Functions for which $\mathcal{F}g = f \ast f$

Suppose one is given $f \in L^{2}(\mathbb{R})$, my question is whether or not there exists a $g \in L^{1}(\mathbb{R})$ such that $f \ast f = \mathcal{F}g$ where $\mathcal{F}$ is the Fourier transform. ...
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### Prove limit of function

Let $f:(a,+\infty) \to \mathbb{R}$ and on every finite $(a,b)$ interval function $f$ is bounded. Then $$\lim_{x \to \infty}\frac{f(x)}{x}=\lim_{x \to \infty}f(x+1)-f(x)$$ How can we prove or ...
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### A continuously differentiable function with vanishing determinant is non-injective?

(This question relates to my incomplete answer at http://math.stackexchange.com/a/892212/168832.) Is the following true (for all n)? "If $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is continuously ...
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Let $m$ is the standard Lebesgue measure and let $g \in L^{1}(dm)$. We know that $m$ is a Radon measure. Is $\mu$ defined by $d\mu = g\, dm$ also a Radon measure? We first claim that $\mu(K) < ... 1answer 32 views ### Dense subset of$L^{2}$such that$x^{-1/2}f \in L^{1}$and$\int_{[0, 1]}x^{-1/2}f\, dx = 0$Does there exist a dense set of functions$f \in L^{2}([0, 1])$such that$x^{-1/2}f(x) \in L^{1}([0, 1])$and$\int_{0}^{1}x^{-1/2}f(x)\, dx = 0$? I've noticed that$\int_{0}^{1}x^{-1/2}f(x)\, dx = ...
This question concerns distributions, as often encountered in PDE theory, which are defined as continuous linear functionals on the space $C_0^{\infty}(\Omega)$ of test functions. The product of two ...
### Consider the mapping $\mathbb{R}^3 (x,y,z)$ to $\mathbb{R}^3(u,v,w)$ given in coordinates
Consider the mapping $\mathbb{R}^3 (x,y,z)$ to $\mathbb{R}^3(u,v,w)$ given in coordinates $$\left\{ \begin{array}{c} u = yz\sin(x)\\ v = y^2 - x\\ w = xz \end{array} \right.$$ Determine the ranks ...