Intereting Posts

When is the Composite with Cube Root Smooth
Finding the probability density function of $Y=e^X$, where $X$ is standard normal
(Why) is topology nonfirstorderizable?
What is the probability that $X<Y$?
Prove $e^{a_1}+…+e^{a_n}\leqslant e^{b_1}+…+e^{b_n}$, where $a_1+…+a_n=b_1+…+b_n=0$, $|a_j|\leqslant |b_j|$
Truth Table for If P then Q
The locus of two perpendicular tangents to a given ellipse
Hard inequality $ (xy+yz+zx)\left(\frac{1}{(x+y)^2}+\frac{1}{(y+z)^2}+\frac{1}{(z+x)^2}\right)\ge\frac{9}{4} $
Prove that if the equation $x^{2} \equiv a\pmod{pq}$ has any solutions, then it has four solutions.
Intersection of nested closed bounded convex sets in Euclidean space
L2 Matrix Norm Upper Bound in terms of Bounds of its Column
Intuitive understanding of the Reidemeister-Schreier Theorem
Closed form of $\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$
If $f: \mathbb{C} \to \mathbb{C}$ is continuous and analytic off $$ then is entire.
For every irrational $\alpha$, the set $\{a+b\alpha: a,b\in \mathbb{Z}\}$ is dense in $\mathbb R$

On $\mathbb{R}^n$ and $p\ge 1$ the $p$-norm is defined as $$\|x\|_p=\left ( \sum _{j=1} ^n |x_j| ^p \right ) ^{1/p}$$

and there is the $\infty$-norm which is $\|x\|_\infty=\max _j |x_j|$. It’s called the $\infty$ norm because it is the limit of $\|\cdot\|_p$ for $p\to \infty$.

Now we can use the definition above for $p<1$ as well and define a $p$-“norm” for these $p$. The triangle inequality is not satisfied, but I will use the term “norm” nonetheless. For $p\to 0$ the limit of $\|x\|_p$ is obviously $\infty$ if there are at least two nonzero entries in $x$, but if we use the following modified definition

$$\|x\|_p=\left ( \frac{1}{n} \sum _{j=1} ^n |x_j| ^p \right ) ^{1/p}$$

then this should have a limit for $p\to 0$, which should be called 0-norm. What is this limit?

- Uniform Integrbility
- Understanding of nowhere dense sets
- Is it possible for a function to be in $L^p$ for only one $p$?
- Continuous with compact support implies uniform continuity
- Convergence of $\sum_{m\text{ is composite}}\frac{1}{m}$
- Show $\sum\limits_{k=1}^{\infty} \frac {1}{(p+k)^2} = -\int_0^1 \frac{x^p \log x}{1-x}\,dx$ holds

- Convergence in $L^\infty$ is nearly uniform convergence
- Fubini's Theorem for Infinite series
- Questions about $f: \mathbb{R} \rightarrow \mathbb{R}$ with bounded derivative
- Self-Contained Proof that $\sum\limits_{n=1}^{\infty} \frac1{n^p}$ Converges for $p > 1$
- Simplification / Concentration Results / Bounds for the Expectation of the Minimum of Two Correlated Random Variables
- Smallest function whose inverse converges
- Construct a continuous monotone function $f$ on $\mathbb{R}$ that is not constant on any segment but $f'(x)=0$ a.e.
- A series involving the harmonic numbers : $\sum_{n=1}^{\infty}\frac{H_n}{n^3}$
- upper limit of $\cos (x^2)-\cos (x+1)^2$ is $2$
- Uniform convergence for $\sum_{n=1}^\infty n^\alpha x^{2n} (1-x)^2$

When $p$ is small, $$x^p = \exp(p \log x) \approx 1 + p\log x.$$ Therefore $$\frac{1}{n} \sum_{j=1}^n x_j^p \approx 1 + p\frac{1}{n} \sum_{j=1}^n \log x_j = 1 + p \log \sqrt[n]{\prod_{i=1}^n x_j}.$$ On the other hand, we have $$(1+py)^{1/p} \longrightarrow \exp(y),$$ and so we easily get that the norm approaches the geometric mean, as Raskolnikov commented.

- Why are polynomials defined to be “formal”?
- Proofs of the properties of Jacobi symbol
- Does existence of anti-derivative imply integrability?
- How is the Integral of $\int_a^bf(x)~dx=\int_a^bf(a+b-x)~dx$
- Definition of hyperbolic lenght.
- Why can the complex conjugate of a variable be treated as a constant when differentiating with respect to that variable?
- How to define a well-order on $\mathbb R$?
- When is $2^n -7$ a perfect square?
- How to solve ~(P → Q) : P & ~Q by natural deduction?
- Connections in non-Riemannian geometry
- Texts on Principal Bundles, Characteristic Classes, Intro to 4-manifolds / Gauge Theory
- Proof the maximum function $\max(x,y) = \frac {x +y +|x-y|} {2}$
- Is $^\omega$ homeomorphic to $D^\omega$?
- Grassman formula for vector space dimensions
- Counting matrices over $\mathbb{Z}/2\mathbb{Z}$ with conditions on rows and columns