pointwise convergence of cdf's, proof of arcsine law for wiener process from...
Let $X_1,X_2,...$ be iid random variables $P(X_i=1)=\frac{1}{2}=P(X_i=-1)$ and let $S_n=X_1+...+X_n$ be simple random walk. Let $M_{2n}$ be the first time of hitting maximum in the walk of $2n$ steps....
View ArticleIntersection of an uncountable set and an uncountable dense set
Question :Let $A,B\subseteq\mathbb{R}$ with $A$ being uncountable and $B$ being uncountable and dense.Must it be true that $A\cap B≠\emptyset?$Since...
View ArticleGiven $f \in L^p_{\text{loc}}$ and $\phi \in S$ (Schwartz class), do we have...
Context. Consider the usual Lebesgue measure on $\mathbb R^n$ and let $L^p_{\operatorname{loc}}$ denote the space of measurable functions that are locally $p$-integrable on $\mathbb R^n$, with $1...
View ArticleGeneralized helicoid is a regular surface.
I'm trying to show that a generalized helicoid is a regular surface, namely that the trace of the function$$x(s,u) = (f(s) \cos u, f(s) \sin u, g(s) + cu),$$where $\delta(s) = (f(s), g(s))$, $s \in...
View ArticleScalar complex holomorphic function - same derivative as scalar real function?
My question is about the derivative of holomorphic complex functions.Assume there is a function $f(x) := \mathbb{R} \rightarrow \mathbb{R}$ , and a function $f(z) := \mathbb{C} \rightarrow \mathbb{C}$....
View ArticleLink between homeomorphism and empty interior [duplicate]
I consider a set $X\subset\mathbb{R}^d$ and an homeomorphism $g : X\to Y\subset\mathbb{R}^d$. I woul like to prove that if $Y$ has non empty interior (for the topology of $\mathbb{R}^d$) then $X$...
View ArticleFinding all differentiable $f: [0,+\infty) \rightarrow [0,+\infty)$ such that...
After some investigation it seems fairly obvious to me that the only such function is the zero function, however I haven't been able to prove it. By considering $$\alpha =\sup\{x\in[0,+\infty) :f(x) =...
View ArticleWhat is the Euclidean norm of the vector containing all $k$-order partial...
Denote $|x|$ the Euclidean norm of a vector $x\in\mathbb R^N$. Also denote $D^kf$ as the vector in $\mathbb R^{N^k}$ containing all $k$-order partial derivatives of the function $f\colon\mathbb...
View ArticleIf $\phi \in C_c^\infty(\mathbb R^n) \cap L^1(\mathbb R^n)$ such that $\|...
Context. Consider the usual Lebesgue measure on $\mathbb R^n$ and denote by $L^1(\mathbb R^n)$ the usual space of measurable functions that are integrable. Moreover, let $C^k(\mathbb R^n)$ denote the...
View ArticleClosed form for this generalisation of the gamma function. $f(x+1)=f(x)g(x+1) $
Just for curiosity I want to generalise the Pi function i.e $f(x+1) = f(x)g(x+1)$ for some differentiable function, I know this function probably has no closed form for general functions $g$ as I...
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Prove that $|z^2+1|\le 2$ implies $|z^3+3z+2|\le 6$
Show that $$\{z \in \mathbb{C}: |z^2+1|\le 2 \} \subseteq \{z \in \mathbb{C} : |z^3+3z+2|\le 6 \} \tag{*}$$(In other words: Let $z\in \mathbb{C}$ satisfy $|z^2+1|\le 2$. Prove that $|z^3+3z+2|\le...
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Question about the Uniqueness of representation theorem for open sets on...
This is from the book $<<$Fundamentals of Real Analysis$>>$ by Sterling K Berberian, Section 5.7 F.Riesz's "Rising Sun Lemma".The author first proves a lemma (The author calls it Riesz's...
View ArticleContinuity of monomial
I want to show formally that a simple monomial $p : \mathbb{R}^n \rightarrow \mathbb{R}$ where $p(x) = x_1\cdots x_n$ is continuous. It should be an easy problem but I can't figure out where to go from...
View ArticleLayer cake representation and function with compact support
My question is related to the posts here and here, but my setup is slightly different.For a real-valued random variable $X$, and a function $\varphi: \mathbb{R} \to \mathbb{R}$ that has support in some...
View ArticleRatio of Noncentral Chi Squared
Let $X$ and $Y$ be independent random variables of noncentral chi squared distributions (https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution)$\chi_a^2(\delta)$ and $\chi_b^2(\eta)$...
View ArticleProve $fg$ is integrable if $f$ and $g$ are both integrable.
Note that for all statements of "integrability" below I am referring to Reimann integrability.Please read: I have already seen the proof where you first prove that $f^2$ is integrable. This I...
View ArticleIs it true that |X| vanishes at infinity, then X is integrable?
Suppose $X$ is a random variable on $(\Omega,\mathscr{F},\mathbb{P})$. If there exists $M>0$, such that for all $\lambda>0$, we have $\mathbb{P}[|X|>\lambda]\le\frac{M}{\lambda}$, then is it...
View ArticleApproximating $\#\{\pi \in S_n:\exists k,\pi(k)=k,~k\leq m\}$
Consider the permutations from $S_n$ and let$$K(n,m)=\#\{\pi \in S_n:\exists k,\pi(k)=k,~k\leq m\}$$be the number of such permutations which have a fixed point in the first $m$ positions. I am...
View ArticleEvaluate $\int\frac1{1+x^n}dx$ for $n\in\mathbb R$
I was wondering on how to evaluate the following indefinite integral for all $n\in\mathbb R$.$$\int\frac1{1+x^n}dx$$It seems to be peculiar in that we...
View ArticleProving that $f(x) = (2^x-1)^{1/x}+ (2^x+3^x-1)^{1/x}$ is increasing for $x...
I want to show that the function $f: [1, \infty) \rightarrow \mathbb{R}$ given by $$f(x) = (2^x-1)^{1/x}+ (2^x+3^x-1)^{1/x}$$ is increasing for $x \in [1, 2]$. By plotting it this seems to be true,...
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