0
votes
2answers
34 views

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 ...
4
votes
0answers
25 views

Consider the sequence $f_n(x) = (\sin(πnx))^n , n = 1, 2, …,$ on the interval $[0,1].$

Consider the sequence $f_n(x) = (\sin(πnx))^n , n = 1, 2, ...,$ on the interval $[0,1].$ Prove that for any $δ > 0$ there is a set $E ⊂ [0,1]$ with $m(E) > 1−δ,$ and a subsequence $f_{n_k} (x), ...
0
votes
0answers
8 views

Does symmetric decreasing rearrangement of a smooth function preserves smoothness?

Let $A\subset \mathbb{R}^n$ a Borel set of finite Lebesgue measure. They define $A^*$ to be the ball centered at 0 with the same measure that $A$. The symmetric-decreasing rearrangement of ...
2
votes
1answer
29 views

composition of $L^{p}$ functions

Suppose $f, g\in L^{p}(\mathbb R), (1\leq p < \infty).$ For simplicity, let us assume that, $g,f:\mathbb R\to \mathbb R$ so that composition of $f$ and $g$, namely, $f\circ g(x)= f(g(x)); (x\in ...
0
votes
0answers
37 views

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 ...
1
vote
5answers
40 views

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 ...
0
votes
1answer
23 views

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 ...
0
votes
1answer
25 views

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 $ ...
2
votes
1answer
43 views

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. ...
2
votes
2answers
93 views

Limit of $S(x) = x − x^2 + x^4 − x^8 + x^{16} − x^{32} + \cdots$ as $x$ approached $1$ from below

I have read the following (http://www.math.harvard.edu/~elkies/Misc/sol8.html) but I dont understand the last part of the solution: For positive $x<1$, consider the alternating sum $$S(x) = x − ...
1
vote
0answers
47 views

existence of a positive root

Consider the polynomial $$ P(\omega)=\omega^8+\phi_7\omega^7+\phi_6\omega^6+\phi_5\omega^5+\phi_4\omega^4+\phi_3\omega^3+\phi_2\omega^2+\phi_1\omega+\phi_0 $$ with real coefficients. Assuming that ...
4
votes
2answers
88 views

Why is there a 'missing' $1$ in the Euler–Mascheroni constant?

It is easy to show that: $$ \sum_{k=1}^n \frac{1}{k} > \ln(n+1), $$ but the Euler–Mascheroni constant is defined as: $$ \gamma = \lim_{n \to \infty} \left( \sum_{k=1}^n \frac{1}{k} - \ln(n) ...
2
votes
0answers
44 views

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 ...
4
votes
1answer
37 views

Mixed partial derivatives are different

Let $f: \Bbb R^2 \to \Bbb R$ be defined as $$f(x) = \left\{ \begin{matrix} x_1^2 \operatorname{arctan} \left( \frac{x_2}{x_1} \right) - x_2^2 \operatorname{arctan} \left( \frac{x_1}{x_2} \right), ...
1
vote
0answers
64 views
+50

Is the upper limit projection Borel

Let $M$ space metric compact, $\pi:M\times\mathbb{R}^k\rightarrow M$ projection such that $\pi(x,y)=x$. Let $f_n:M\times\mathbb{R}^k\rightarrow \mathbb{R}$ continuous and ...
1
vote
0answers
20 views

Equivalent optimization problems?

I am wondering if the set of optimizers of the problem $$ \min_{x \in X} \ f(x) \quad \text{subject to: } g(x) \leq 0, \ h(x) = 1 $$ is the same of the one of $$ \min_{x \in X} \ f(x) + h(x) \quad ...
1
vote
1answer
31 views

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 $L$, ...
0
votes
0answers
15 views

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
1
vote
1answer
35 views

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 ...
1
vote
2answers
33 views

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 $g \in L_{p'}(E)$, ...
1
vote
0answers
31 views

Function satisfies differential equation.

Given the D'Alembert operator D'Alembertian $\Box$, I want to show that $$ G(x,t,x_0,t_0):= \frac{\delta \left(t_0 + \frac{||x-x_0||}{c} -t \right)}{||x-x_0||} $$ satisfies $$ \Box G(x,t,x_0,t_0) = ...
0
votes
1answer
34 views

Proving differentiability using Caratheodory's Lemma

Let $I$ be an open interval and let $c\in I$. Let $f:I\rightarrow\mathbb{R}$ be continuous and define $g:I\rightarrow\mathbb{R}$ by $g(x)=\left|f(x)\right|$. Prove that if $g$ is ...
2
votes
0answers
31 views

rewriting the inverse image

If $\phi_k:\mathbb{R}^2\rightarrow \mathbb{R}$ are continuous functions, for all $k\geq0$ and $$\phi=\limsup_{n\rightarrow \infty }\phi_n$$ Let $A\subset \mathbb{R}$, is possible to write ...
0
votes
0answers
16 views

Constant solutions of separable ODE

Consider the IWP $$ y'(x) = g(x) \cdot h(y(x)), \quad y(x_0) = y_0 $$ for continuous functions $g : I \to \mathbb R$ and $h : U \to \mathbb R$ on open intervals $I, U$ with $(x_0, y_0) \in I\times ...
5
votes
1answer
131 views

Prove that $\displaystyle \lim_{n \rightarrow \infty} \frac{1}{n}\int^n_0xf(x)dx = 0$

Let $f(x) \geq 0$ be continuous on the interval $[0, \infty)$, and suppose that $\int_0^\infty f(x)dx < \infty$. Prove that $\displaystyle \lim_{n \rightarrow \infty} \frac{1}{n}\int^n_0xf(x)dx = ...
2
votes
1answer
36 views

Suppose that the infinite series $\displaystyle \sum_{n=1}^\infty \mu\{x \in X : |f_n(x)| \geq \epsilon\}$ converges for each $\epsilon > 0.$

Let $\{f_n\}$ be a sequence of measurable functions on a measure space $(X, \mathcal{M}, \mu)$. Suppose that the infinite series $\displaystyle \sum_{n=1}^\infty \mu\{x \in X : |f_n(x)| \geq ...
0
votes
0answers
24 views

Fixed point of projected operator

Let $X \subset \mathbb{R}^n$ be a compact convex set and let $f: X \rightarrow X$ be Lipschitz continuous and such that $$ \left( f(x) - f(y) \right)^\top \left( x-y\right) \leq 0 $$ for all $x, y ...
1
vote
1answer
30 views

Convex combinations sequence question

Let $a_0,a_1,\beta$ be given with $0<\beta<1$ Let the sequence be defined by $a_{n+2} = \beta a_{n+1} + (1-\beta)a_n$ for $n\geq0$ Show that $\{a_n\}$ converges and find its limit.. How to ...
1
vote
1answer
20 views

Does existence of global minimum imply coercivity?

It is known that a coercive function over a closed, unbounded set has a global minimum. Is the converse true ? The larger context for this question is the following question: Suppose we are given a ...
0
votes
1answer
39 views

Linearizing systems about critical points.

$$\def\q{\begin{pmatrix}}\def\p{\end{pmatrix}}\def\l{\lambda}\def\f{\frac{\sqrt{11}}{2}}$$ Find all the critical points of the following systems and derive the linearised system about each ...
0
votes
0answers
19 views

Laplacian in arbitrary spherical coordinates?

assume that $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is a function that just depends on the radius vector $r$, so no angular dependence(!). Can we say what $\Delta f$ is in arbitrary coordinates $r ...
0
votes
1answer
70 views
+200

Product of two $2$-variables Taylor series

Using the standard multi-index notation, suppose we have the two Taylor series $$ f(\theta) := \sum_{|\alpha|=0}^{\infty} a_{\alpha} \theta^{\alpha} $$ and $$ g(\theta) := \sum_{|\alpha|=0}^{\infty} ...
1
vote
1answer
37 views

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 ...
1
vote
0answers
121 views
+50

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 ...
2
votes
1answer
38 views

Prove that for any $1 < p < ∞$ there exists a function $f ∈ L_p(μ)$ such that $f \notin L_q(μ)$ for any $q > p.$

Let $(X, Ω, μ)$ be a finite measure space. Assume that for any $t > 0$ there exists $E ∈ Ω$ satisfying $0 < μ(E) < t.$ Prove that for any $1 < p < ∞$ there exists a function $f ∈ ...
0
votes
1answer
59 views
+50

Monotonic Functions and Uniform Convergence

The following is a proof from "Heavy-Tail Phenomena" by Resnick (2007). I have some questions about the proof. (2.3) seems to be an identity. The left side the global sup over $[a, b]$ and hence ...
3
votes
2answers
75 views

Showing a set is nowhere dense in $C([0,1])$

Let $E_n$ be the set of all $f \in C\big([0,1]\big)$ for which there exists $x_0 \in [0,1]$ (depending on $f$) such that \begin{align*} \lvert\, f(x)-f(x_0)\rvert \leq n\lvert x-x_0\rvert, ...
1
vote
4answers
68 views

How to prove that $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}\mathrm dx$ exists

I am trying to show that the integral $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}\mathrm dx$ exists ($n$ is a natural number). I tried to use the comparison theorem by bounding from above the ...
0
votes
1answer
38 views

What is missing? (Rudin's Principles of Mathematical Analysis - Theorem 2.30)

Let us first give a definition: Definition Given a metric space $X$, and a subset $Y\subseteq X$, we say a subset $E$ of $Y$ is open relative to $Y$ if for each $p\in E$ there is an associated ...
2
votes
0answers
57 views

simultaneous trigonometric equations

Consider the pair of simultaneous equations $p_5\cos(2\omega\tau)+p_4\omega\sin(2\omega\tau) = p_1\omega^2-p_3-p_6\cos(3\omega\tau) $, $p_4\omega\cos(2\omega\tau)-p_5\sin(2\omega\tau) = ...
0
votes
1answer
18 views

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 ...
1
vote
1answer
16 views

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
2
votes
2answers
49 views

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 ...
3
votes
1answer
71 views

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. ...
0
votes
1answer
37 views

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 ...
2
votes
2answers
105 views

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 ...
0
votes
0answers
17 views

Is $d\mu = g\, dm$ a Radon measure for $g \in L^{1}(dm)$?

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) < ...
0
votes
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 = ...
1
vote
1answer
23 views

Products of distributions ill-defined

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 ...
1
vote
1answer
55 views

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 ...