Does Cantor's set contain a copy of each finite set?
Let $C$ denote the (usual) Cantor's set in the interval $[0,1]$. If $S=\{x_1,x_2,\cdots,x_n\}$ is a finite set of points in $\mathbb R$, is it true that $aS+b\subset C$ for some $a\neq 0$ and...
View ArticleSchauder estimate for $f \in L^\infty$
I was reading an article where at some point the author uses the following estimate:Let $u$ be a solution of$$\Delta u = f \quad \text{in } B_1$$for $f \in L^\infty$. Then $u \in C^{1,1 - \varepsilon}$...
View ArticleHow to prove the following identity [closed]
Let $f$ be a function continous on closed bounded interval $[a,b]$How to prove$$\lim_{h \to 0^+} \frac{1}{h} \int_{a}^{a+h} f(x) dx = f(a)$$Only using continuity of $f$ and properties of lebesque...
View ArticleWrite real number $x>1$ as product of terms of the form $(1+1/p)$ where $p$...
If $x>1$, then there is a sequence of primes $p_1, p_2, ...$ (finite or infinite) so that $x=\prod_i(1+1/p_i)$. How one can prove this? Is there a method to find such sequense (In the finite case)?...
View ArticleConstructing Lebesgue measure recursively
I am trying to work out what I think (if successful) is the "most natural" construction of Lebesgue measure, using transfinite recursion. There are already multiple papers on this, including two papers...
View ArticleMaclaurin second degree approximation of natural exponential function for...
Could there be any reason why Maclaurin second degree approximation is being considered for approximating Exp(x) in optimizing average cost in inventory model? Many scholars use it in their articles.
View ArticleStieltjes Integration Problem
Consider a non-negative bounded function $f:[-R,R]\rightarrow\mathbb R$ with infimum bounded away from $0$, and the function $g:[-R,R]\rightarrow\mathbb R$ with $g(x) = x - \operatorname{sgn}(x)R$....
View ArticleWhy does the graph of an integrable function have volume zero?
Let $A\subset \mathbb R^m$ be a rectangle and $f:A\to \mathbb R$ be a bounded integrable function. I'm trying to prove the graph of $f$ has volume zero.We define the volume of a J-measurable set $X$ as...
View ArticleRank of a matrix with entries 0 or 1 [closed]
Let $A$ be an $n\times n$ matrix over the finite field $\mathbb{F}_2$, whose diagonal entries are $0$ and all others are $1$. Then what is the rank of $A$?
View ArticleA special integral with a squared dilogarithm
The following problem is proposed by Cornel Ioan Valean, that is, to prove that$$\int_0^{\pi/2} (\operatorname{Li}_2(\sin^2(x))^2\textrm{d}x=\int_0^{\pi/2}...
View ArticleSimple conditions making $\phi^{-1}(\frac{1}{n}\phi(x))$ concave
Let $\phi:(0,\infty)\to(-\infty, b)$ twice differentiable such that:$\phi$ strictly concave and strictly increasing$\phi(1)=0$, $\phi'(1)=0$$\lim_{y\to0} \phi(y)=-\infty$Then the convex, strictly...
View ArticleJensen's inequality with affine combination [closed]
From page 217 of 'Convex functions' by Arthur Wayne Roberts, Dale Varberg (exercise F) i want proof that: $f \mathbb{R}\to\mathbb{R}$ is affine iff $ f(\sum_{i=1}^n\lambda_ix_i)\le...
View ArticleThe set of Rational Numbers : Q doesn't have the least upper bound property
In the following proof why do we have to take k1 to be in integers and not in the natural numbers since it is already established that 1<x<2Second, How we do we know for sure that k1 is a natural...
View ArticleArea of a Rectangle From the Definition of Area Found in a Multivariable...
The book is Multivariable Calculus by Peter Lax and Maria Terrell. First some background.Let $D$ be a bounded region in the plane and let $h>0$. Divide the whole plane into squaresof side length $h$...
View ArticleInterval related to increasing/decreasing and concavity/convexity
Why do some people use closed intervals when describing the intervals where a function is increasing/decreasing or concave/convex?For example, given the function $f(x)= x^2-5x+6$, it says the interval...
View ArticleSome guidance needed (Spivak Chapter 5 Problem 20)
If $f(x)=x$ for rational $x$, and $f(x)=-x$ for irrational $x$,then$\lim_{x \to a} f(x)$ does not exist for any $a\neq 0$**Proof:**Intuitively, it makes sense that the limit exists at $0$ because $x$...
View ArticleShow that $ Y $ is separable, and that the spectrum of $ A$ consists of...
Let $A: X \to Y$ be a linear operator defined on a dense subspace $ X $ of a Banach space $ Y $. Assume that $ A $ has an inverse that is compact as a linear operator on $ Y $. Show that $ Y $ is...
View ArticleSteiner symmetrization of functions in two dimensions
Given a smooth function $u$ defined on $\mathbb{R}^2$, restrict $u$ to a subset $\Omega \subset \mathbb{R}^2$ (possibly not simply connected) foliated by level sets of a smooth function $\psi: \Omega...
View ArticleI am unable to solve this question any hint will be helpful [closed]
Consider the vector space $V:=\{f\,\colon \mathbb{R} \to \mathbb{R}\, \colon f \text{ is a polynomial}\}$.If$$W =\{f \in V \colon f(m+1) + f(m-1)-2f'(m)=0 \text{ for every integer }\, m, |m|\leq...
View ArticleShow that $T$ is bounded if $fT$ is linear functional
Let $X$ and $Y$ be Banach space. If $T: X\to Y$ is a linear map so that $f(T)\in X^*$ (dual space of $X$) for every $f\in Y^*$, then $T$ is bounded.Here is the proof by contradiction of this question,...
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