— In the present paper, we prove the finiteness of the set of moderate rational points of a once-punctured elliptic curve over a number field. This finitenessmay be regarded as an analogue for a once-punctured elliptic curve of the well-known finiteness of the set of torsion rational points of an abelian variety over a number field. In order to obtain the finiteness, we discuss the center of th...

we introduce a two generated weakly branch contracting automaton group $g$ which is generated by a two state automaton on a three letter alphabet. using its branch structure and the finiteness nature of a sequence of its factor groups we compute the order of some of these factors. furthermore some algebraic properties of $g$ are detected .

Gorenstein homological dimensions are refinements of the classical homological dimensions, and finiteness singles out modules with amenable properties reflecting those of modules over Gorenstein rings. As opposed to their classical counterparts, these dimensions rarely come with practical and robust criteria for finiteness, even over commutative noetherian local rings. Indeed, over such a ring ...

Kierstead showed that every computable marriage problem has a computable matching under the assumption of computable expanding Hall condition and computable local finiteness for boys and girls. The strength of the marriage theorem reaches WKL0 or ACA0 if computable expanding Hall condition or computable local finiteness for girls is weakened. In contrast, the provability of the marriage theorem...

Let Rn = max0≤j≤n Sj − Sn be a random walk Sn reflected in its maximum. We give necessary and sufficient conditions for finiteness of passage times of Rn above horizontal or certain curved (power law) boundaries. Necessary and sufficient conditions are also given for the finiteness of the expected passage time of Rn above linear and square root boundaries. 2000 MSC Subject Classifications: prim...

This paper addresses a decision problem highlighted by Grigorchuk, Nekrashevich, and Sushchanskĭı, namely the finiteness problem for automaton (semi)groups. For semigroups, we give an effective sufficient but not necessary condition for finiteness and, for groups, an effective necessary but not sufficient condition. The efficiency of the new criteria is demonstrated by testing all Mealy automat...

In the study of substitutative dynamical systems and Pisot number systems, an algebraic condition, which we call ‘weak finiteness’, plays a fundamental role. It is expected that all Pisot numbers would have this property. In this paper, we prove some basic facts about ‘weak finiteness’. We show that this property is valid for cubic Pisot units and for Pisot numbers of higher degree under a domi...

Anosov representations give a higher-rank analogue of convex cocompactness in rank-one Lie group which shares many its good geometric and dynamical properties; finiteness rank one may be seen as controlled weakening to allow for isolated failures hyperbolicity. We introduce relatively dominated relativization representations, or other words finiteness. prove that groups admitting must hyperboli...

The residual finiteness of the class of groups of fibred knots, or those knot groups with finitely generated and, therefore, free commutator subgroups, has been known for some time. Using Baumslag's results on absolutely parafree groups, this paper extends the result to twist knots (Whitehead doubles of the trivial knot) and certain other classes of nonfibred knots whose minimal spanning surfac...

The residual finiteness of the class of groups of fibred knots, or those knot groups with finitely generated and, therefore, free commutator subgroups, has been known for some time. Using Baumslag's results on absolutely parafree groups, this paper extends the result to twist knots (Whitehead doubles of the trivial knot) and certain other classes of nonfibred knots whose minimal spanning surfac...