Articles of analysis

Where do these p-adic identities come from?

I was reading this article (http://www.asiapacific-mathnews.com/03/0304/0001_0006.pdf) to see some applications of $p$-adic numbers outside mathematics, and came across these two identities: $\sum_{n=0}^\infty (-1)^n n!(n+2) = 1$ and $\sum_{n=0}^\infty (-1)^n n!(n^2-5) = -3$. Since no $p$ was stated, I figured there must be some trick that shows these are the values in any $\mathbb{Q}_p$, but I […]

Let $(f_n)$ be a sequence of continuous functions from the reals to the reals. And let $\sup{|f_n(x)-f_m(x)|} \leq \frac1{\min(m,n)}$

Let $(f_n)$ be a sequence of continuous functions from the reals to the reals. And let $\sup{|f_n(x)-f_m(x)|} \leq \frac{1}{\min(m,n)}$ Prove there is a continuous functions $f$ such that $$\lim_{n\rightarrow \infty} \sup{|f_n(x)-f(x)|}=0$$ Start: I think this is an $\epsilon/3$ proof. Let $\epsilon >0$ and let $a$ be a real number. Since each $f_n$ is continuous, there […]

A question on second symmetric derivatives

Suppose function $f:J→R$ on an open interval $J$ satisfies following inequality: $$\left|f(x)+f(y)−2f\Big(\frac{x+y}{2}\Big)\right|\leq C(f)\, |x−y|^{1+\delta}$$ for all $x,y∈J$, and for some $\delta \in (0,1].$ Is follow from this fact that $f\in C^{1}$ or $C^{1+\delta}$?

Find an equation on the cone which is tangent plane is perpendicular to a given plane

Let $z=\sqrt{x^2+y^2}$ be cone. Find an equation of each plane tangent to the cone which şs perpendicular to the plane $x+z=5$ I have learnt the solution of such question for parallel at previous question I asked today. But now, I want to learn properly this questions for perpendicular. Again, I have its answer. But I […]

Numerical integration over a surface of a sphere

I am integrating a double integral in spherical coordinates over the surface of a sphere in MATLAB numerically. Although I have changed the relative and absolute tolerance I get the feeling that this algorithm never terminates. And when I checked the values of my function that MATLAB had evaluated everything looked fine, no huge oscillations, […]

$S\subset \mathbb{R}^2$ with one and only one limit point in $\mathbb{R}^2$ such that no three points in $S$ are collinear

Question as in the title, but here it is re-typed just in case not all of the title is visible on your screen (you’re welcome): I am interested if there is a set $S\subset \mathbb{R}^2$ with one and only one limit point in $\mathbb{R}^2$ such that no three points in $S$ are collinear. I also […]

Criteria for positive semi-definiteness – zero diagonal

I am currently doing a bit of background reading on some fundamental topics in preparation for a talk, and came across a question relating to positive definiteness. It is taken from Horn and Johnson’s book entitled Matrix Analysis, and reads as follows: Show that if a positive semidefinite matrix has a zero entry on the […]

simultaneous trigonometric equations

Consider the pair of simultaneous equations $p_5\cos(2\omega\tau)+p_4\omega\sin(2\omega\tau) = p_1\omega^2-p_3-p_6\cos(3\omega\tau) $, $p_4\omega\cos(2\omega\tau)-p_5\sin(2\omega\tau) = \omega^3-p_2\omega+p_6\sin(3\omega\tau)$. Forget about all the other parameters/variables in the equations. All I want is to find a closed form expression for $\tau$. Thanks for your help and ideas!

Proving that $f(z) = \frac{1}{2} \left(z + \frac{1}{z}\right)$ is biholomorph on a certain set

Let $f: \mathbb{C} \backslash \{0\} \to \mathbb{C}$ be given by $$f(z) := \frac{1}{2} \left(z + \frac{1}{z}\right)$$ I first want to find the image of the set $H = \{z \in \mathbb{C}: |z| < 1, Im(z) > 0\}$ (the open upper half of the unit circle) regarding $f$. Next, I want to show that $f$ sends […]

are there $x_{1},x_{2} \in $ such that $x_{1}-x_{2}=1$ and $f(x_{1})=f(x_{2})$?

This question already has an answer here: Universal Chord Theorem 4 answers