Positive and negative portions of conditionally convergent series
I want to prove that for a conditionally convergent series $\sum_{n = 1}^{\infty} a_n$, that its positive subseries (Let $K := \{k \vert a_k > 0\}$, and the positive subseries is $\sum_{j \in K}...
View ArticleHow can any supremum of a proper subset of the real numbers be outside the...
Let E ⊂ R, E is bounded above, and α = sup E. Then ∀ε> 0, there exists some t ∈ E such thatα−ε< t ⇒α− t < εSince α> t, α− t = |α− t|. It follows that|α− t| < εwhich implies α = t ∈ E.Why...
View ArticleAsymptotics: How to simplify the term $s_0 = \alpha_1 n^{\beta_1} + \alpha_2...
Assume $s_0 = \alpha_1 n^{\beta_1} + \alpha_2 n^{\beta_2} \pm n^{\beta_1-1-\varepsilon}$, with $\alpha_1, \alpha_2 > 0$ and $\beta_1 > \beta_2$ and $\varepsilon > 0$. Can we deduce that $s_0 =...
View ArticleDoes lebesgue measurable function not lebesgue integrable on every closed...
I am trying to solve this exercise:Let $f$ be a lebesgue measurable on the line. if for any $r \in \mathbb{Q}$, $f(x+r) = f(x)$$ (a.e. x)$. Prove that there exist a constant $C$, such that $f = C...
View ArticleIs it possible to get a lower bound of mollified function with Sobolev norm?
Let $f\in C_c(\mathbb{R}^n)$ and $\rho_\epsilon$ be a mollifier with support in $B(0,\epsilon)$. Is it possible to get a lower bound on $||\rho_\epsilon*f||_{H^s}$ in terms of $||f||_{H^s}$ if one can...
View ArticleClosed, bounded and noncompact set
I am referring to Exercise 7.4.17 from the book Basic Analysis I by Lebl. I was only able to find this, but it is a proof verification using a somewhat different approach.Let $(X,d)$ be an incomplete...
View ArticleGeneralized of Miklos Schweitzer 1980 P1
question; For a real number $ x$, let $ \|x \|$ denote the distance between $ x$ and the closest integer. Let $ 0 \leq x_n <1 \; (n=1,2,\ldots)\ ,$ and let $ \varepsilon >0$. Show that there...
View ArticleTranslation of odd and even functions
Let $\varphi: \mathbb{R} \to \mathbb{R}$ be a periodic function of period $L>0$, that is,\begin{equation}\label{periodicitycondition}\varphi(x+L)=\varphi(x),\; \forall\; x \in \mathbb{R}....
View ArticleCalculation of the volume of a solid of rotation
Calculate the centroid of the homogeneous solid generated by the rotation around the y-axis of the domain in the xy-plane defined by:$$D=\left \{(x,y)\in \Bbb R^2 : x \in [1,2],0\leq y \leq...
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Convergence of sequences formed by orthocenters, incenters, and centroids in...
Now asked on MO here.Given a scalene triangle $A_1B_1C_1$ , construct a triangle $A_{n+1}B_{n+1}C_{n+1}$ from the triangle $A_nB_nC_n$ where $A_{n+1}$ is the orthocenter of $A_nB_nC_n$, $B_{n+1}$ is...
View ArticleShowing Set of Increasing Sequences of Natural numbers $\{n_1
Showing the set of increasing(strictly) sequences of natural numbers $\{n_1<n_2<\cdots\}$has the same cardinality as set of real numbers of the form $0.\alpha_1\alpha_2\cdots$----Zorich...
View Article(Infinite) sum of products of 'offset' Bessel functions (an application to...
TLDR version: I'm trying to calculate $\sum_{n=1}^{\infty} J_{n-1}(a) J_{-n-1}(a)$ but can't seem to get it right. For motivation & attempt, read the rest.Consider the harmonic "FM" oscillator...
View ArticleProof of Kolmogorov's Existence Theorem in Dudley's Real Analysis and...
The proof is written allowing for a state space that is an arbitrary measurable space, but I am interested in the case of real valued stochastic processes.I am working through the proof of the theorem,...
View ArticleCurve discussion for a real function with a real parameter
Consider the function\begin{eqnarray*}f_{a}(z)=\left(1-a\right)\cdot \frac{z\cdot \left(\tanh\left(a\cdot z\right)+\tanh\left(z\right)\right)}{\ln\left(\cosh\left(a\cdot...
View ArticleShow that the given map is continuous in a Hausdorff space
Let $E$ is an $n$-dimensional Hausdorff topological vector space and $F$ is any topological vector space. If, $\{e_1, \cdots,e_n \}$ any basis for $E$ and $T: E \to F$ is linear map, then show...
View ArticleAlgebraic proof of $\tan x>x$
I'm looking for a non-calculus proof of the statement that $\tan x>x$ on $(0,\pi/2)$, meaning "not using derivatives or integrals." (The calculus proof: if $f(x)=\tan x-x$ then $f'(x)=\sec^2...
View ArticleHeat Equation : Can a singularity develop away from origin?
Consider the following mixed problem for a radial, nonlinear, 2D-heat equation$$\begin{cases}u_t = u_{rr} + \dfrac{1}{r}u_r + F(t,r,u,u_r), \quad (t,r) \in [0,+\infty) \times [0,1] \\u(0,r) = f(r) \in...
View ArticleThe Fourier transform of $e^{-i/x}$
$\def\R{\mathbb R}$Question.Does anyone know what is the Fourier transform of$$ f(x)=e^{-i/x} $$on the real line? I would like to compute it explicitly, or to establish some properties to have a good...
View ArticleProve existence of solution for IVP
I am trying to solve this ODE problem. I am studying for an exam.Let $f \colon \mathbb{R}^n \rightarrow \mathbb{R}^n$ be $C^1$ such that $|f(x)| \leq 1 + |x|^{\alpha}$. Consider the...
View ArticleQuestion about partial derivative of logarithm
Let $\Omega$ be an open set and $f \in H(\Omega)$. I want to compute$\frac{\partial \log(f\overline{f})}{\partial \overline{z}}$.What I've tried:$\frac{\partial \log(f\overline{f})}{\partial...
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