2,114,214 research outputs found
Rational points on X_0^+ (p^r)
We show how the recent isogeny bounds due to \'E. Gaudron and G. R\'emond
allow to obtain the triviality of X_0^+ (p^r)(Q), for r>1 and p a prime
exceeding 2.10^{11}. This includes the case of the curves X_split (p). We then
prove, with the help of computer calculations, that the same holds true for p
in the range 10 < p < 10^{14}, p\neq 13. The combination of those results
completes the qualitative study of such sets of rational points undertook in
previous papers, with the exception of p=13.Comment: 16 pages, no figur
On -null sequences and their relatives
Let and , where is the
conjugate index of . We prove an omnibus theorem, which provides numerous
equivalences for a sequence in a Banach space to be a -null
sequence. One of them is that is -null if and only if is
null and relatively -compact. This equivalence is known in the "limit"
case when , the case of the -null sequence and -compactness.
Our approach is more direct and easier than those applied for the proof of the
latter result. We apply it also to characterize the unconditional and weak
versions of -null sequences
Characterization of tropical hemispaces by (P,R)-decompositions
We consider tropical hemispaces, defined as tropically convex sets whose
complements are also tropically convex, and tropical semispaces, defined as
maximal tropically convex sets not containing a given point. We introduce the
concept of -decomposition. This yields (to our knowledge) a new kind of
representation of tropically convex sets extending the classical idea of
representing convex sets by means of extreme points and rays. We characterize
tropical hemispaces as tropically convex sets that admit a (P,R)-decomposition
of certain kind. In this characterization, with each tropical hemispace we
associate a matrix with coefficients in the completed tropical semifield,
satisfying an extended rank-one condition. Our proof techniques are based on
homogenization (lifting a convex set to a cone), and the relation between
tropical hemispaces and semispaces.Comment: 29 pages, 3 figure
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