Can discrete time make continuous space look discrete?

Van Bendegem has recently offered an argument to the effect that, if time is discrete, then there should exist a correspondence between the motions of massive bodies and a discrete geometry. On this basis, he concludes that, even if space is continuous, it should nonetheless appear discrete. This paper examines the two possible ways of making sense of that correspondence, and shows that in neither case van Bendegem's conclusion logically follows

Topological equivalence and rigidity of flows on certain solvmanifolds

Given a Lie group $G$ and a lattice $\Gamma$ in $G$, a one-parameter subgroup $\phi$ of $G$ is said to be rigid if for any other one-parameter subgroup $\psi$, the flows induced by $\phi$ and $\psi$ on $\Gamma\backslash G$ (by right translations) are topologically orbit-equivalent only if they are affinely orbit-equivalent. It was previously known that if $G$ is a simply connected solvable Lie group such that all the eigenvalues of $\mathrm{Ad} (g)$, $g\in G$, are real, then all one-parameter subgroups of $G$ are rigid for any lattice in $G$. Here we consider a complementary case, in which the eigenvalues of $\mathrm{Ad} (g)$, $g\in G$, form the unit circle of complex numbers.Let $G$ be the semidirect product $N \rtimes M$, where $M$ and $N$ are finite-dimensional real vector spaces and where the action of $M$ on the normal subgroup $N$ is such that the center of $G$ is a lattice in $M$. We prove that there is a generic class of abelian lattices $\Gamma$ in $G$ such that any semisimple one-parameter subgroup $\phi$ (namely $\phi$ such that $\mathrm{Ad} (\phi_t)$ is diagonalizable over the complex numbers for all $t$) is rigid for $\Gamma$ (see Theorem 1.4). We also show that, on the other hand, there are fairly high-dimensional spaces of abelian lattices for which some semisimple $\phi$ are not rigid (see Corollary 4.3); further, there are non-rigid semisimple $\phi$ for which the induced flow is ergodic.</jats:p

Topological equivalence and rigidity of flows on certain solvmanifolds

Given a Lie group $G$ and a lattice $\Gamma$ in $G$, a one-parameter subgroup $\phi$ of $G$ is said to be rigid if for any other one-parameter subgroup $\psi$, the flows induced by $\phi$ and $\psi$ on $\Gamma\backslash G$ (by right translations) are topologically orbit-equivalent only if they are affinely orbit-equivalent. It was previously known that if $G$ is a simply connected solvable Lie group such that all the eigenvalues of $\mathrm{Ad} (g)$, $g\in G$, are real, then all one-parameter subgroups of $G$ are rigid for any lattice in $G$. Here we consider a complementary case, in which the eigenvalues of $\mathrm{Ad} (g)$, $g\in G$, form the unit circle of complex numbers. Let $G$ be the semidirect product $N \rtimes M$, where $M$ and $N$ are finite-dimensional real vector spaces and where the action of $M$ on the normal subgroup $N$ is such that the center of $G$ is a lattice in $M$. We prove that there is a generic class of abelian lattices $\Gamma$ in $G$ such that any semisimple one-parameter subgroup $\phi$ (namely $\phi$ such that $\mathrm{Ad} (\phi_t)$ is diagonalizable over the complex numbers for all $t$) is rigid for $\Gamma$ (see Theorem 1.4). We also show that, on the other hand, there are fairly high-dimensional spaces of abelian lattices for which some semisimple $\phi$ are not rigid (see Corollary 4.3); further, there are non-rigid semisimple $\phi$ for which the induced flow is ergodic

Posterior Fossa Neurenteric Cysts Can Expand Rapidly: Case Report

Neurenteric cysts are considered congenital lesions that may slowly expand over time. Although more commonly found in the spinal canal, they may be found intracranially, particularly in the posterior fossa. Here, we present an unusual case of a large, rapidly expanding histologically confirmed posterior fossa neurenteric cyst in a 53-year-old woman, who presented with quadriparesis. Computed tomography imaging done ~1.5 years before admission failed to demonstrate any obvious abnormality; however, the lesion had grown to 4 cm in maximal dimension at presentation with significant mass effect. The lesion was resected microsurgically using a retrosigmoid approach. The patient improved postoperatively and was neurologically intact at last follow-up. We could find no other documented case of marked, rapid expansion of a neurenteric cyst in the literature. We conclude that, although neurenteric cysts are thought to be congenital, they can undergo rapid expansion even in adults. The mechanism of expansion is unknown but may involve increased secretion, hemorrhage, or inflammation. We discuss the surgical management and review the literature in view of this surprising finding

Posterior fossa neurenteric cysts can expand rapidly: case report.

Neurenteric cysts are considered congenital lesions that may slowly expand over time. Although more commonly found in the spinal canal, they may be found intracranially, particularly in the posterior fossa. Here, we present an unusual case of a large, rapidly expanding histologically confirmed posterior fossa neurenteric cyst in a 53-year-old woman, who presented with quadriparesis. Computed tomography imaging done ~1.5 years before admission failed to demonstrate any obvious abnormality; however, the lesion had grown to 4 cm in maximal dimension at presentation with significant mass effect. The lesion was resected microsurgically using a retrosigmoid approach. The patient improved postoperatively and was neurologically intact at last follow-up. We could find no other documented case of marked, rapid expansion of a neurenteric cyst in the literature. We conclude that, although neurenteric cysts are thought to be congenital, they can undergo rapid expansion even in adults. The mechanism of expansion is unknown but may involve increased secretion, hemorrhage, or inflammation. We discuss the surgical management and review the literature in view of this surprising finding