How to solve a series of integral equation [closed]
I am trying to solve the following integral equation analytically:$$\sum_{n \geq 1} \left( \int_0^\tau e^{-n^2(\tau-s)} f_n(s) \, ds \right) = g(\tau), \quad \tau \in [0, T],$$where $(f_n(\tau))_n$ is...
View ArticleWhere is the mistake in this line integral evaluation?
Consider the vector function$$ \vec{f}(x,y) = \begin{bmatrix} y \\ x \end{bmatrix} \tag{1} $$The goal is to evaluate the line integral$$ \int_{(0,0)}^{(a,b)} \vec{f}(x,y)\cdot d\vec{x} \tag{2} $$with...
View ArticleProve that the limit of a sequence of finite measures that is uniformly...
I am having difficulty solving an exercise in the book "Modern Real Analysis" by Ziemer. It is exercise 7 of section 6 on signed measures in the chapter on integration (chapter 6). The exercise...
View ArticleAsymptotic of a trigonometric product
Consider the product $P_n(x) = \sin(x-a)\sin(2x-a)\cdots\sin(2^n x - a)$, where $a \in [0,\pi/2]$.What is the asymptotic of $\max |P_n(x)|$. More precisely, is there a positive constant$A = A(a)$, such...
View ArticleSpivak Chapter 7, Problem-15
If $\phi$ is continuous and $\lim_{x \to \infty} \frac{\phi(x)}{x^n}=0=\lim_{x \to -\infty} \frac{\phi(x)}{x^n}$ then(a) Prove that if $n$ is odd then there is a number $x$ such that...
View ArticleHow to prove this inequality $\left|x\sin{\frac{1}{x}}-y\sin{\frac{1}{y}}\right|
For any real numbers $x,y\neq 0$,show that$$\left|x\sin{\dfrac{1}{x}}-y\sin{\dfrac{1}{y}}\right|<2\sqrt{|x-y|}$$I found this problem when I dealt with this problem. But I can't prove it. Maybe the...
View ArticleContinuity of optimal value of a functional on a Hilbert space
Let $f:X\times Y\to \mathbb{R}$ be a bounded continuous function on the product topological space $X\times Y$. It is well known that if $Y$ is compact, then the infimum function $g(x) = \inf_{y\in...
View ArticleSubtlety within understanding proof of First Order Condition of Quasiconvex...
Given $f: \mathbb{R} \to \mathbb{R}^n$ is differentiable and $dom f$ is convex, we must prove$$f \ \ \text{is quasiconvex iff} \\f(y) \le f(x) \implies \nabla{f}^T(x)(y-x) \le 0$$ for all $x,y \in dom...
View ArticleProving density of the rationals in the reals
Theorem: $a,b \in \mathbb{R}: b>a \implies \exists q \in \mathbb{Q}: a < q < b $Proof attempt:We need to show that we can find integers $m,n$ such that $a < \frac mn < b$ or $na < m...
View ArticleThe equality case of the Schwartz inequality
Question:The fact that $a^2 \geq 0$ $ \forall a \in \mathbb{R}$; elementary as it may seem, isnevertheless the fundamental idea upon which most important inequalitiesare ultimately based. The...
View ArticleConvergent Sequences Satisfying a Specific Functional Equation
Let $\alpha, \beta$ be real numbers. Find all convergent sequences $(a_n)_{n \geq 1}$ satisfying$$\alpha(a_1 + \cdots + a_n) + \beta(a_1 \cdot \ldots \cdot a_n) = 1$$for all $n \geq 1$.AttemptNote that...
View ArticleInterpretation of Complex Differentiability for Functions from $\mathbb{R}^2$...
I understand that any complex-valued function $ f(z) = u(x, y) + iv(x, y) $, where $ z = x + iy $, can be viewed as a function from $ \mathbb{R}^2 $ to $ \mathbb{R}^2 $ by considering $ f(z) $ as the...
View ArticleVolume of a box fromed by countable disjoint boxes
ProblemLet $n>1$. Suppose\begin{equation}\prod_{j=1}^n [a_j,b_j)=\bigsqcup_{k=1}^{\infty} \prod_{j=1}^n [a_{j,k},b_{j,k})\end{equation}The symbol $\bigsqcup$ means disjoint union.Prove the following...
View ArticleOn mollifiers acting between $L^2$ and Sobolev spaces
Consider a sequence of finite lattices in $\mathbb{R}^n$ defined by$$L_k= [-k,k]^n \cap 2^{-k}\cdot \mathbb{Z}^n, \quad k=1,2,3,\ldots$$Write $H^s(\mathbb{R}^n)$ for the Sobolev space of exponent $s$...
View ArticleSum of two function sequences [closed]
Let (fn) and (gn) be two function sequences that converge uniformly to the functions f and g respectively. Let a and b two real numbers.Then the function sequence (a.fn + b.gn) converges uniformly to...
View ArticleThe class of functions preserving measurability of functions in post composition
Is it possible to characterize the class of functions $g:{\bf R}\rightarrow {\bf R}$ which satisfies the following condition: for every measurable function $u:{\bf R}\rightarrow {\bf R}$ it follows...
View ArticleIs this set closed, despite being union of an open and a closed set?
I'm perhaps losing myself in a glass of dirty water but I have a weird doubt. Is the set $(0, 1) \cup \mathbb{N}$ closed?I mean if I had $[0, 1] \cup \mathbb{N}$, then I would know that it's closed...
View ArticleSpivak Exercise 3-39: Proving Change of Variables using Sard's Theorem
$\DeclareMathOperator{\ext}{ext}$In Exercise 3-39 of Spivak's Calculus on Manifolds, Spivak asks the reader to use Sard's Theorem to prove the following (which is his original change of variables...
View ArticleLet $V$ be a bounded open subset of $\mathbb{R}^n.$ Is $B(0,1)$ the union of...
Let $V$ be a bounded open subset of $\mathbb{R}^n,$ and define $B(x,r)$ to be the open ball with centre $x$ and radius $r.$True or false: $B(0,1)$ is the union of countably many pairwise disjoint...
View Article$f \in C^{\infty} (\mathbb{R}^3, \mathbb{R}^4)$,...
We have $f \in C^{\infty} (\mathbb{R}^3, \mathbb{R}^4)$, $f = f_1, f_2, f_3, f_4$.Elements of that transformation satisfy:$$f_1(x)^2 + f_2(x)^2 = 1 = f_3(x)^2 + f_4(x)^2 \ \text{ for: } \ x \in...
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