Normalized hamming distance in probability
Let two functions $f,g :[n]\to \{0,1\},$ then we define $$\delta{(f,g)}=\frac{|\{i\in[n]:f(i)\neq g(i) \}|}{n}$$ is called normalized hamming distance.My teacher said me this $\delta{(f,g)}$ is...
View ArticleIs $\mathbb{R}$-{transcendental numbers} still a field or not? [duplicate]
Is $\mathbb{R}$-{transcendental numbers} still a field or not?i.e,can we prove finite many terms of + and $\times$ using non-transcendental numbers isstill non-transcendental numbers or...
View ArticleUnderstanding the proof of $L^{\infty}$ is complete.
I got lost when reading the proof of $L^{\infty}$ is complete. The book proceed the proof as follows:We show that each absolutely convergent series in $L^{\infty}(X,\mathscr{A},\mu)$ is convergent. We...
View ArticleDoes it make sense to talk about regularity of boundaries in 1D?
The following is a definition from Evans' PDE book:Let $U \subset \mathbb{R}^n$ be open and bounded, $k = \{1, 2, \ldots, \}$. We say that a boundary $\partial U$ is $C^k$ if for each point $x^0 \in...
View ArticleTwo series converges pointwise to same limit, and one converges uniformly
We have $\sum{f_n}$ and $\sum{g_n}$.$f_n,g_n$:[0,1]->$\mathbb{R}$ and all f are non-negative and all g are continuous. Both series converges pointwise to the same limit. How to prove that if $\sum...
View ArticleProve that for the given enumeration of rationals and a sequence of positive...
Here is the question:Let $\left\{q_1, q_2, \ldots\right\}$ be an enumeration of all rational numbers in the interval $[0,1]$. This means that $q_k$'s are rational numbers in $[0,1]$ and that every...
View ArticleAbsolutely continuous function and continuously differentiability
It’s easy to prove that if $ f \in C^1([a,b], \mathbb{R})$ than $ f \in AC([a,b], \mathbb{R})$.I was wandering what if $ f \in C^1((a,b), \mathbb{R})$. I know that it is still true, for example, if $||...
View ArticleShow that f is periodic if $f(x+a)+f(x+b)=\frac{f(2x)}{2}$?
Suppose $a$ and $b$ are distinct real numbers and $f$ is a continuous real function such that $\frac{f(x)}{x^2}$ goes to 0 when $x$ goes to infinity or minus infinity. Suppose that$...
View ArticleLimit of function whose inverse has limit approaching infinity at finite point
Consider a continuous strictly monotone function $f:(-1,1)\to\mathbb{R}$, where$$\lim_{x\to1}f(x)=\infty, \lim_{x\to-1}f(x)=-\infty$$I have proved that this function is bijective. I want to prove that...
View ArticleBarycentre of a ball
I saw the following definition for the barycentre of a set $\Omega \subseteq \mathbb{R}^d$:$$\mathrm{bc}^\Omega=\frac{1}{\mathrm{vol}(\Omega)}\int_\Omega x dx \in\mathbb{R}^n.$$So I wanted to compute...
View ArticleA question of convex function with out $C^1$ condition.
It is a confusing question.Set $f:[0,+\infty)\to \mathbb{R}$ is a strictly convex downward function satisfy $$\lim _{x\to +\infty}\frac{f(x)}{x}=+\infty,$$ prove that $\int_0^\infty \sin f(x)dx$ is...
View ArticleMultiplication of convergent sequence and multiplication of series
It is a basic rule that $((\lim_{n\to \infty}a_{n}=A\in\mathbb{R})\wedge (lim_{n\to \infty }b_{n}=B\in\mathbb{R})\Rightarrow lim_{n\to\infty}(a_{n}b_{n})=AB)$And according to the definition of...
View ArticleRiemann integrability for step function
Here is the problem:Let $f(x):=2$ if $0\le x<1$ and $f(x):=1$ if $1\le x\le 2$. Show that $f\in\mathcal{R}[0,2]$ and evaluate its integral.Here is my working so far:Fix $c\in\mathbb{R}$ and define...
View Article$\mathbb R^n$ version of proof that uniform continuity implies boundedness...
I have seen a lot of questions on the site about the proof that if $f$ is uniformly continuous on an interval, then its range is bounded. However, I have not been able to find a question that addresses...
View ArticleConditions that a sequence should satisfy to be an eventually monotone sequence
Let $\{a_n\}_{n\in\mathbb{N}}$ be a sequence of real numbers such that:$a_n\in[0,1]$, $\forall n\in\mathbb{N}$$\lim_{n\to\infty}a_n = 0$$\lim_{n\to\infty}\frac{a_{n+1}}{a_n} = 1$$a_{n+1} \leq a_n$,...
View ArticleProof of Inequality Involving Ordered Real Numbers and Geometric Means
Let $x_1, x_2, \dots, x_n$ be real numbers such that $x_n \geq x_{n-1} \geq \dots \geq x_1$ and $0 < x_i \leq \frac{1}{n}$, for any $i$. Define $t = \left(x_1 \cdot x_2 \cdot \dots \cdot...
View ArticleShowing sequence converges knowing its bounded and $a_{n+1} - a_n...
Let $(a_n)$ be bounded sequence, s.t for all $n \in \mathbb{N}$, $a_{n+1}\geq a_{n} - \dfrac{1}{2^n}$, I'm stuck on showing $(a_n)$ converges.
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Is the feasible region for convex combinations of powers of $N$ cosines...
Let $x = \sum_{i=1}^N a_i \cos^2(\theta_i)$ and $y = \sum_{i=1}^N a_i \cos^3(\theta_i)$ be the two convex combinations of powers of cosines, where the variables satisfy the following constraints(1)....
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Why $W$ is open in baby Rudin 9.28
9.28 Theorem: Let $f$ be a r $\mathscr{C}'$ -mapping of an open set $E$$ \subset \mathbb{R}^{n+m}$ into $\mathbb{R}^n$, suchthat $f(a, b) = 0$ for some point $(a, b) \in E$.Put $A = f'(a, b)$ and...
View ArticleHow to prove : If $f:(a,b)\to\mathbb{R}$ is strictly increasing then...
How can we prove the statement: if $f:(a,b)\to\mathbb{R}$ is strictly increasing then $\lim_{x\to b}f(x)\neq-\infty$I think I was able to prove this adding an extra restriction that $f$ is continuous,...
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