Cardinality of the set of functions defined on a finite set .
If $F$ be the set of all functions defined on $I_n =\{ 1 , 2 , 3 ,..., n\} , n\in \mathbb{N} $ with range $B\subseteq I^+$( set of positive integers ) . Then(a) $F$ is countable(b) $F$ is uncountable...
View ArticleIf $f$ is Lipschitz, $X_n$ converges in distribution and $d(X_n,Y_n)$...
Let$(E,d)$ be a separable metric space$X,X_n,Y_n$ be random variables with values in $E$ such that $(X_n)_{n\in\mathbb{N}}$ converges in distribution to $X$ and...
View ArticleCalculate the normal cone to $X=\{f: [0,1] \rightarrow [0,1]\mid f \text{...
Let $X:=\{f: [0,1] \rightarrow [0,1]\mid f \text{ increasing}\} $. We endow it with $L_2$ norm, and thus $X \subset L_2([0,1])$.We can show that the set $X$ is closed and thus compact, and also...
View ArticleThe condition "mass preserving maps" in the motivation of optimal transportation
I am reading Monge's formulation of the optimal transportation problem. It says that one wishes to find the a transport map $T: (X,\mu) \rightarrow (Y,\nu)$ that minimizes the transport cost $$\int_X...
View ArticlePair of real functions satisfying some conditions [closed]
Consider two functions $\psi$ and $\varphi$ defined on the interval $(0,c)$ where $c\in(0,+\infty)$ and they exhibit the following characteristics:$\psi$ and $\varphi$ are continuous, positive, and...
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Inequality involving entropies: $\left \|p -\frac{1}{n} e \right \|_2\ge\left...
For a given probability vector $p=(p_1,\dots,p_n)$ with $p_1,\dots,p_n > 0, \sum_{i=1}^n p_i=1$, I want to prove the following inequality$$\left \|p -\frac{1}{n} e \right \|^2_2=\sum_{i=1}^n \left...
View ArticleCurious interchange of the order of summation
Usually interchanging the order of summation requires $\sum_n\sum_m|a_{n,m}|<\infty$. Unfortunately, I don't have this condition on my hands. Only conditions I have are:$$\sum_n\left|\sum_m...
View ArticleHomework Question related to Poisson Kernel
The following is a question from my homework:Let $a_n$, $b_n$ be the Fourier coefficient of a $2\pi$-periodic, integrable function $f$ on $[-\pi$,$\pi$]. ($a_n$ is the Fourier coefficient of $cos(nx)$...
View ArticleConcavity of function implies convex upper contour
Today I saw a theorem in class that stated the following:$f$ is concave $ \Rightarrow\{z \in \mathbb R^n : f(z) \ge c\}$ is convex.The proof is relatively straight forward and I understand. However, I...
View ArticleProve $\lim_{x \to 1} \frac{1}{1+x} = \frac{1}{2}$ with $ \varepsilon -...
$\lim\limits_{x \to 1} \frac{1}{1+x} = \frac{1}{2}$Here is my attemptPrep work:$$\Bigl| \frac{1}{1+x} - \frac{1}{2} \Bigr| = \frac{1}{2} \Bigl| \frac{1-x}{1+x}\Bigr|=\frac{1}{2}\cdot |x-1| \cdot...
View ArticleProve $\lim_{x \to 1} \sqrt{x^2 + 1} = \sqrt{2}$ with $\varepsilon - \delta$
$\lim\limits_{x \to 1} \sqrt{x^2 + 1} = \sqrt{2}$My attemptPreliminary work:$$| \sqrt{x^2 + 1} - \sqrt 2 | = \Bigl| \frac{(\sqrt{x^2 + 1} - \sqrt 2)(\sqrt{x^2 + 1} + \sqrt 2)}{\sqrt{x^2 + 1} + \sqrt 2}...
View ArticleEvaluate...
I know some basic properties of the Dirac delta 'function', and I'm familiar with the sifting property. In any case, I'm stuck trying to evaluate this double...
View ArticleBounded, measurable and supported on a set of finite measure function
Suppose f is a bounded and measurable function on R and supported on a set of finite measure. Prove that for every $\epsilon \gt 0$ there exists a simple function $s$ such that $\int |f-s|dx$ $\lt...
View ArticleHow to show that the increment operation for natural numbers is well defined?
Let $a,b$ be natural numbers and $++$ be an increment operation. Based on Peano axioms alone, how to show that if $a=b$, then $a++ = b++$?Do note that the book I am reading (Real Analysis by Terrence...
View ArticleAlternative proof that there is no natural number between two consecutive...
First the book defines the reals as a set with two operations that satisfy the field and order axioms. Then, a set $S \subseteq \mathbb{R}$ is said to be inductive if $1 \in S$ and if $x \in S \implies...
View ArticleA basic property of slowly varying functions
It seems that a fundamental property of a slowly varying function is that for all $\delta > 0$$$ \lim_{x\to \infty} L(x) \, x^{-\delta} = 0.$$How to prove this? The only book I found (that was free)...
View ArticleShowing that $\lim\sup s_n = \lim\inf s_n =s$
I have the following problem:Let $s$ be a real number and $(s_n)$ be a sequence of real numbers. Suppose that for any subsequence $(s_{n_{k}})$ of $(s_n)$, $(s_{n_{k}})$ has a subsequence...
View ArticleIntegral $\int^{\pi/2}_{0} \left(...
I need to prove the following relation:$$\int^{\pi/2}_{0} \left( \int^{\pi/2}_{0}f(1-{\sin\theta}{\cos\phi})\sin\theta \,\, d\theta \right) d\phi= {\pi/2}\int^1_0f(x)\,\,dx$$My guess is that this...
View ArticleSome maximal inequality of Fefferman-Stein type
Let $1 < p < \infty$. Fefferman-Stein shows thatthere exists a constant $c_p > 0$ such that\begin{equation} \int_{\mathbb{R}^n} (M[f](x))^p \phi(x) \,dx \leq c_p \int_{\mathbb{R}^n} (f(x))^p...
View ArticleLimit superior proof doubt
There is a question about proving the Limit superior theorem. It starts with the "Let $(a_n)$ be a bounded sequence. Prove that the sequence defined by $y_n = \sup\{a_k : k ≥ n\}$ converges." The...
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