Computational complexity of real numbers
Recently, I've been studying computable analysis. One of the basic notions is a computable real number, which I will define as any $r \in \mathbb{R}$ which has a computable Cauchy name - a computable,...
View ArticleSolve the differential equation $f'=f/(f\circ f)$
I am completely stuck on the following differential equation:Find all differentiable functions $ f:\mathbb{R}_{>0}\to\mathbb{R}_{>0} $ such that:$$ f' = \frac{f}{f\circ f}. $$The identity...
View ArticleNecessary and sufficient conditions for $\|f\|_p = \|f\|_q$ with $p \neq q$
Let $0 < p < q \leq \infty$ and suppose $ E\subset \mathbb{R}^N$ with $m(E)=1$ (where $m$ is the Lebesgue measure). I am asked to find necessary and sufficient conditions for:$$ \left(...
View ArticleDoes a closed subalgebra of $C(X)$ which separates points and vanishes...
Let $X$ be a compact Hausdorff space and $C(X)$ be the real valued continuous functions on $X$. Suppose that $\mathcal A$ is a (topologically) closed subalgebra of $C(X)$. Is it true that if $\mathcal...
View ArticleConfusion between exponential integral and incomplete gamma function
In its page on exponential integralHERE, Wikipedia assumes that $E_n(x)$ can be written as a special case of the upper incomplete gamma function:$$E_n(x)=x^{n-1}\Gamma(1-n,x)$$I suppose that $n \in...
View ArticleProving that $\lim_{n \to \infty} \sum_{k=0}^{n} \binom{n}{k}...
I need to prove the following limit:$$\lim_{n \to \infty} \sum_{k=0}^{n} \binom{n}{k} \frac{(-1)^k}{k!} = 0.$$Attempts:I tried using generating functions and combinatorial identities, but I couldn't...
View ArticleProve convergence in probability for this sequence
Consider a sequence $z_1, z_2, \dots$ of real numbers, defined by: $$z_n := n^{-1}\sum_{k=1}^n \operatorname E\left[\vert c_kX_k\vert^2\cdot I_{A_{n,k,\varepsilon}}\right],$$ where $X_1, X_2, \dots,...
View ArticleProve that every subset of $\mathbb{R}$ is compact in the finite complement...
I need help with my proof in particular. I am aware that there is a similar question elsewhere. Can someone please verify my proof or offer suggestions for improvement?Prove that every subset of...
View Article6 team round robin play over 8 nights, 3 games played consecutively per...
What is a possible schedule for a 6-team round-robin tournament played over 8 days with 3 games per day, totaling 24 total games, ensuring each team plays the last game on an equal number of days?The...
View ArticleConvergence of $\sum(u_n)$ if $\frac{u_{n+1}}{u_n} \leq \left ( \frac{n}{n+1}...
I got this question:let $\alpha >1$, and let $u_{n}$ be a sequence of positive numbers such that for all n:$$ \frac{u_{n+1}}{u_n}\leq \left ( \frac{n}{n+1} \right )^{\alpha}. $$I have to prove that...
View ArticleConditions for making a function with Fourier transform to an absolutely...
There are many absolutely non-integrable functions with convergent Fourier transform, i.e., $$X(f) = \int_{-\infty}^{+\infty}x(t)\exp(-2\pi ift)dt \ \ \text{converges but }...
View ArticleIs there a sequence $(a_n)$ so that for every $r \in \mathbb{R}$, there is a...
My guess is that there is no such sequence and prove it by contradiction.My attempt is as follows: Let $A_r$ denote the set containing all terms of such subsequence for each $r \in \mathbb{R}$. Since...
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The Fubini-Tonelli Theorem for Complete Measures (Problem Folland 2.49).
I'm working on proving the Fubini-Tonelli theorem for complete measures in Folland's real analysis text. The theorem reads as followsThe exercise is:49. Prove Theorem 2.39 by using Theorem 2.37 [the...
View ArticleApproximation by regular curves
Let $\gamma:[0,1]\to\mathbb{R}^d$ be a smooth curve. We say $\gamma$ is regular iff $\gamma^\prime $ does not have any zeros. Can we approximate any smooth curve $\gamma$ uniformly by regular curves...
View ArticleHalf Range Sine Series
Question:It is known that $f(x)=(x−4)^2$ for all $x\in [0,4]$.Compute the half range sine series expansion for $f(x)$.My answer :Half range series: $p=8$, $l=4$,...
View ArticleProve that $\lim_{x \to \frac{1}{2}} \left( \frac{1}{x^3 + 1} \right) =...
I am trying to show that the given limit exists by use of the epsilon-delta definition. Below is my working thus far:Let $f(x) = \frac{1}{x^3 + 1}$. If $\lim_{x \to \infty} = \frac{8}{9}$, then forany...
View ArticleSectorial Operator
let $\mathcal{H}$ be a hilbert space and $A : \mbox{dom } A \subseteq \mathcal{H} \to \mathcal{H}$ a linear operator. Furthermore for $\omega \in [0,\pi]$ let\begin{equation}S(\omega) := \begin{cases}...
View ArticleConstraints on a Function Mapping a distribution to Itself
I have a function $f: \mathbb{R}^d \to \mathbb{R}^d$ smooth and invertible defined as $y = f(x)$. The input $x$ has independent components following a uniform distribution:$$x_i \sim \mathcal{U}[-0.5,...
View ArticleUnder what assumptions on f can I calculate f([a, b])?
I know that:If $f$ is continuous and monotonous on $[a, b]$, then $$f([a, b]) = [\min(f(a), f(b)), \max(f(a), f(b))]$$If $f$ is continuous and unimodal on $[a, b]$ with an extremum at $x_0 \in [a, b]$,...
View ArticleSelf Study - Folland Real Analysis Problem(2.1.3)
Currently I am self-studying Measure Theory before studying Measure-Theoretic Probability Theory. The following is a problem I solved recently from Folland's Real Analysis, in particular...
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