Articles of reference request

Lower bound on convexity radius in terms of injectivity radius (without using curvature)

Let $M$ be a complete Riemannian manifold, and let $C$ be a subset of $M$. We will say $C$ is convex if for any points $p,q \in C$, there exists a unique normal minimal geodesic $\gamma$ joining $p$ and $q$, such that $\gamma \subset C$. We will define: the convexity radius of $M$ by “the […]

A question about an asymptotic formula

I’ve been told that the asymptotic formula $\pi(x+y)-\pi(x)\sim y/\ln x$ holds for $y\ge x^{1/2+\varepsilon}$ if Riemann’s hypothesis is true, but I was unable to find a journal reference for this. Does anybody know of any journal reference or any other source where I can find this conditional result?

Is a simple curve which is nulhomotopic the boundary of a surface?

Let $C$ be a simple curve in an open subset $U$ of $\mathbb R^3$. Suppose that $C$ is nulhomotopic in $U$. Must there exist a homeomorphism $f$ from the closed unit disk $D$ in $\mathbb R^2$ to $U$ such that $f(\partial D) = C$? This seems intuitively like it should be true, and I believe […]

Quick question: Direct sum of zero-dimensional subschemes supported at the same point

For a torsion free sheaf $E$ on a surface $X$ we have $0\rightarrow E\rightarrow E^{**}\rightarrow Z\rightarrow 0$ where $Z$ is a zero dimensional subscheme of $X$. Let $p\in\mathbb{P}^2$ be a closed point. Let $\mathcal{I}_p$ be the corresponding ideal sheaf. We have $(\mathcal{I}_p^{\oplus 2})^{**}=\mathcal{O}^{\oplus 2}$. Hence $\mathcal{O}^{\oplus 2}/\mathcal{I}_p^{\oplus 2}$ is a zero dimensional subscheme of $\mathbb{P}^2$. […]

Conditional probability and the disintegration theorem

I was wondering how conditional probability and the disintegration theorem are related? How is the conditional probability given by the disintegration theorem? I don’t quite understand what Wikipedia says: The disintegration theorem can be applied to give a rigorous treatment of conditioning probability distributions in statistics, while avoiding purely abstract formulations of conditional probability. The […]

Is there a name for this family of probability distributions?

I am wondering whether a family of probability distributions with the following form of a density function has a name: $$f(x)=C*\operatorname{Exp}(-B|x|^A)$$ where $A$, $B$ and $C$ are positive constants, with $B$ being a “scaling” constant and $C$ selected so that $f(x)$ integrates to 1. When $A=1$ this corresponds to a Laplace distribution (with $B=1/b$ and […]

Book on coordinate transformations

I am looking for a book that covers various coordinate systems in 3 dimensions, various methods of representing rotations and other transformations like rotation matrices and quarternions, including algorithms for conversions between various coordinate systems and representations of transformations. Is there a single book that covers these.

Greatest common divisor of real analytic functions

Consider two real-valued real analytic functions $f$ and $g$. I want to prove that there exists a greatest common divisor $d$, which is a real analytic function. By greatest common divisor, I mean the following: Common divisor: There exist real analytic functions $q_1, q_2$ such that $f = dq_1, g = dq_2$, and Greateast: If […]

The class of all functions between classes (NBG)

Is it possible in NBG (von Neumann-Bernays-Gödel set theory) to construct the class of all functions $X \to Y$ between two (proper) classes $X,Y$? I guess that this does not work. In the special case $Y=\{0,1\}$ we would get the class of all subclasses of $X$, which does not exist. Can someone confirm this?

where can i find this A.H. Stone's theorem proof?

can someone tell me where can i find a proof of the following theorem (by A.H.Stone) : “an uncountable product of Hausdorff non-compact spaces is never normal ” ? thanks in advance !