Intereting Posts

Limit of $L^p$ norm
definition of “weak convergence in $L^1$”
Integration of $\int_0^\infty\frac{1-\cos x}{x^2(x^2+1)}\,dx$ by means of complex analysis
Contour integral of $\int_0^{2\pi} \frac{1}{A – cos \theta} d\theta$
Is $123456788910111121314\cdots$ a $p$-adic integer?
Asymptotic expansion of the integral $\int_0^1 e^{x^n} dx$ for $n \to \infty$
Prove that there is no element of order $8$ in $SL(2,3)$
Prove that the only sets in $R$ which are both open and closed are the empty set and $R$ itself.
Intuition: Power Set of Intersection/Union (Velleman P77 & Ex 2.3.10, 11)
How to show that the circle group T is isomorphic to $\mathbb R/ \mathbb Z$
Convergence in product topology
What is $dx$ in integration?
Pushforward of Inverse Map around the identity?
When is the union of topologies a topology?
Advice for writing good mathematics?

consider this a soft-question.

Information Theory is fairly young branch of mathematics (60 years).

I am interested in question, whether there have been any information theoretic results that had impact on other seemingly ‘unrelated’ branches of mathematics?

Looking forward to hearing you thoughts.

- How to make the encoding of symbols needs only 1.58496 bits/symbol as carried out in theory?
- What is the trellis diagram for a linear block code?
- Weighing Pool Balls where the number of balls is odd
- what is the mutual information of three variables?
- Lower bound on binomial coefficient
- What is the least amount of questions to find out the number that a person is thinking between 1 to 1000 when they are allowed to lie at most once

- What is the origin of “how the Japanese multiply” / line multiplication?
- History of dot product and cosine
- History of notation: “!”
- How did we know to invent homological algebra?
- Has error correction been “solved”?
- What's the hard part of zero?
- where does the term “integral domain” come from?
- Is there any surprising elementary probability problem that result in surprising solution like the Monty Hall problem?
- What are some examples of proofs using the Pythagorean assumption that all segments are commensurable?
- Volumes of cones, spheres, and cylinders

There is a theorem that no comparison sorting algorithm can have a worst-case running time better than $O(n\log n)$, and this limit is often referred to as an *information-theoretic* limit on the running time. (For example, see Leiserson et al. *Introduction to Algorithms* (2001) pp. 181 and preceding, or Knuth *The Art of Computer Programming* **3** (1998) pp. 182–183 “Relation (1) is often called the *information-theoretic lower bound*…”.)

This is because there are $n!$ possible orders for a list of $n$ items, and therefore $\lceil \log_2 n!\rceil$ bits are required to specify one of these orders. By Stirling’s approximation, this is $O(n\log n)$ bits.

But a comparison of two elements produces only one bit of information about the permutation, so $O(n\log n)$ comparisons are required to identify which permutation is given.

The paper “The information theoretic bound for problems on ordered sets and graphs” (Michael Saks, 1985) says “This is an example of an information theoretic bound (ITB) for a combinatorial decision problem, one of the few general techniques known for obtaining lower bounds on the computation complexity of a problem. … This paper is a survey of a number of computational problems involving ordered sets andgraphis in which the ITB can be applied.”

- Demystify integration of $\int \frac{1}{x} \mathrm dx$
- Find the sum : $\frac{1}{\cos0^\circ\cos1^\circ}+\frac{1}{\cos1^\circ \cos2^\circ} +\frac{1}{\cos2^\circ \cos3^\circ}+…+$
- Prove that there exists a Borel measurable function $h: \mathbb R \to \mathbb R$ such that $g= h\circ f$.
- Proving that $\int_0^1 f(x)e^{nx}\,{\rm d}x = 0$ for all $n\in\mathbb{N}_0$ implies $f(x) = 0$
- Sufficient conditions for embedding a set of $n$ points with a given metric in $\mathbb{R}^n$.
- How to find angle x without calculator?
- Question about computing a Fourier transform of an integral transform related to fractional Brownian motion
- Series $\frac{x^{3n}}{(3n)!} $ find sum using differentiation
- Convergence test of the series $\sum\sin100n$
- How to prove(or disprove) $\begin{vmatrix} A&B\\ B&A \end{vmatrix}=|A^2-B^2|$
- Understanding countable ordinals (as trees, step by step)
- Martingale and Submartingale problem
- How prove $(1-\sum_{i=1}^{n}a^3_{i})^{1/3}\cdot (1-\sum_{i=1}^{n}b^3_{i})^{1/3}\ge 1-\sum_{i=1}^{n}a_{i}b_{i}-\sum_{i=1}^{n}|a_{i}-b_{i}|$
- Existence of a sequence with prescribed limit and satisfying a certain inequality II
- Definition of $L^0$ space