Related papers: A local-global theorem on periodic maps

Let $\psi_1,...,\psi_k$ be periodic maps from $\Bbb Z$ to a field of characteristic p (where p is zero or a prime). Assume that positive integers $n_1,...,n_k$ not divisible by p are their periods respectively. We show that…

Let f be a function from the set of rational numbers into itself. We call f a global power map if f(n) = n^k for some integer exponent k. We call f a local power map at the prime number p if f induces a well-defined group homomorphism on…

Let $G$ be a locally compact group. For every $G$-flow $X$, one can consider the stabilizer map $x \mapsto G_x$, from $X$ to the space $\mathrm{Sub}(G)$ of closed subgroups of $G$. This map is not continuous in general. We prove that if one…

A permanental field, $\psi=\{\psi(\nu),\nu\in {\mathcal{V}}\}$, is a particular stochastic process indexed by a space of measures on a set $S$. It is determined by a kernel $u(x,y)$, $x,y\in S$, that need not be symmetric and is allowed to…

Let ${\mathbb Z}_p$ denote the ring of all $p$-adic integers and call $${\mathcal U}=\{(x_1,\ldots,x_n):\,a_1x_1+\ldots+a_nx_n+b=0\}$$ a hyperplane over ${\mathbb Z}_p^n$, where at least one of $a_1,\ldots,a_n$ is not divisible by $p$. We…

Classical theorems from the early 20th century state that any Haar measurable homomorphism between locally compact groups is continuous. In particular, any Lebesgue-measurable homomorphism $\phi:\mathbb{R} \to \mathbb{R}$ is of the form…

The paper proves two theorems concerning the set of periods of periodic orbits for maps of graphs that are homotopic to the constant map and such that the vertices form a periodic orbit. The first result is that if $v$ is not a divisor of…

Let X be a connected open set in n-dimensional Euclidean space, partially ordered by a closed convex cone K with nonempty interior: y > x if and only if y-x is nonzero and in K. Theorem: If F is a monotone local flow in X whose periodic…

For the delayed logistic equation $x_{n+1} = a x_n (a-x_{n-1})$ it is well known that the nontrivial fixed point is locally stable for $1<a\leq 2$, and unstable for $a>2$. We prove that for $1<a\leq 2$ the fixed point is globally stable, in…

Let $F$ be a global field, let $\vp \in \Fx$ be a rational map of degree at least 2, and let $\a \in F$. We say that $\a $ is periodic if $\vpn (\a) = \a$ for some $n \geq 1$. A Hasse principle is the idea, or hope, that a phenomenon which…

Given a particular collection of categorical axioms, aimed at capturing properties of the category of locales, we show that if $\mathcal{C}$ is a category that satisfies the axioms then so too is the category $[ G, \mathcal{C}]$ of…

Let {\Lambda}\subsetR^{n}\timesR^{m} and k be a positive integer. Let f:R^{n}\rightarrowR^{m} be a locally bounded map such that for each ({\xi},{\eta})\in{\Lambda}, the derivatives D_{{\xi}}^{j}f(x):=|((d^{j})/(dt^{j}))f(x+t{\xi})|_{t=0},…

Let $G=(V,E)$ be a finite graph. For $v\in V$ we denote by $G_v$ the subgraph of $G$ that is induced by $v$'s neighbor set. We say that $G$ is $(a,b)$-regular for $a>b>0$ integers, if $G$ is $a$-regular and $G_v$ is $b$-regular for every…

We show that if the Hessian of a $C^{1,1}$ function has uniformly positive determinant almost everywhere then its index is locally constant, as conjectured by \v{S}ver\'ak in 1992. We deduce this result as a consequence of a more general…

A Boolean function $f:V \to \{-1,1\}$ on the vertex set of a graph $G=(V,E)$ is locally $p$-stable if for every vertex $v$ the proportion of neighbours $w$ of $v$ with $f(v)=f(w)$ is exactly $p$. This notion was introduced by Gross and…

We prove a version of both Jacobi's and Montel's Theorems for the case of continuous functions defined over the field $\mathbb{Q}_p$ of $p$-adic numbers. In particular, we prove that, if \[ \Delta_{h_0}^{m+1}f(x)=0 \ \ \text{for all}…

Let $S$ be a noetherian scheme and $f\colon X\to S$ be a smooth morphism of relative dimension 1. For a locally constant sheaf on the complement of a divisor in $X$ at over $S$, Deligne and Laumon proved that the universal local acyclicity…

Let $E/F$ be a quadratic extension of $p$-adic fields and $\textrm{U}_{2r+1}$ be the unitary group associated with $E/F$. We prove the following local converse theorem for $\textrm{U}_{2r+1}$: given two irreducible generic supercuspidal…

Let $K$ be a number field, $f\in K[x]$ a quadratic polynomial, and $n\in\{1,2,3\}$. We show that if $f$ has a point of period $n$ in every non-archimedean completion of $K$, then $f$ has a point of period $n$ in $K$. For $n\in\{4,5\}$ we…

Given a map $\varphi:\mathbb{P}^1\rightarrow \mathbb{P}^1$ of degree greater than 1 defined over a number field $k$, one can define a map $\varphi_\mathfrak{p}:\mathbb{P}^1(\mathfrak{o}_k/\mathfrak{p})\rightarrow…