Vaught's conjecture and the Glimm-Effros property for Polish transformation groups

We extend the original Glimm-Effros theorem for locally compact groups to a class of Polish groups including the nilpotent ones and those with an invariant metric. For this class we thereby obtain the topological Vaught conjecture.

     

Vaught's conjecture on unitary relations

We confirm the Vaught's conjecture and obtain a complete classification for the theories of unitary relations. Project supported by the National Natural Science Foundation of China     

Proof of the Branner-Hubbard conjecture on Cantor Julia sets

By means of a nested sequence of some critical pieces constructed by Kozlovski, Shen, and van Strien, and by using a covering lemma recently proved by Kahn and Lyubich, we prove that a component of the filled-in Julia set of any polynomial is a point if and only if its forward orbit contains no periodic critical components. It follows immediately that the Julia set of a polynomial is a Cantor set if and only if each critical component of the filled-in Julia set is aperiodic. This result was a conjecture raised by Branner and Hubbard in 1992. This work was supported by the National Natural Science Foundation of China     

Sharper changes in topologies

Let be a Polish group, a Polish topology on a space , acting continuously on , with -invariant and in the Borel algebra generated by . Then there is a larger Polish topology on so that is open with respect to , still acts continuously on , and has a basis consisting of sets that are of the same Borel rank as relative to .

     

A graph-theoretic version of the union-closed sets conjecture

An induced subgraph S of a graph G is called a derived subgraph of G if S contains no isolated vertices. An edge e of G is said to be residual if e occurs in more than half of the derived subgraphs of G. We introduce the conjecture: Every non-empty graph contains a non-residual edge. This conjecture is implied by, but weaker than, the union-closed sets conjecture. We prove that a graph G of order n satisfies this conjecture whenever G satisfies any one of the conditions: δ(G) ≤ 2, log₂ n ≤ δ(G), n ≤ 10, or the girth of G is at least 6. Finally, we show that the union-closed sets conjecture, in its full generality, is equivalent to a similar conjecture about hypergraphs. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 155–163, 1997     

Nonexistence of abelian difference sets: Lander's conjecture for prime power orders

In 1963 Ryser conjectured that there are no circulant Hadamard matrices of order 4$"> and no cyclic difference sets whose order is not coprime to the group order. These conjectures are special cases of Lander's conjecture which asserts that there is no abelian group with a cyclic Sylow -subgroup containing a difference set of order divisible by . We verify Lander's conjecture for all difference sets whose order is a power of a prime greater than 3.

     

Zassenhaus conjecture for A 6

For the alternating group A ₆ of degree 6, Zassenhaus’ conjecture about rational conjugacy of torsion units in integral group rings is confirmed. Dedicated to the memory of I. S. Luthar Professor Luthar died at the age of 74 in December 2006.     

Vaught's Conjecture and Group Rings

We discuss Vaught's conjecture for complete theories of modules over some group rings over the integers and other Dedekind domains, and over pullback rings of Dedekind domains.     

Fibrations et conjecture de Tate

Résumé

Nous décrivons le comportement du rang du groupe de Mordell-Weil de la variété de Picard de la fibre générique d’une fibration en termes de contributions locales données par des moyennes de traces de Frobenius agissant sur les fibres. Les énoncés fournissent une réinterpretation de la conjecture de Tate (pour les diviseurs) et généralisent des résultats antérieurs de Nagao, Rosen-Silverman et des auteurs.

Abstract. Fibrations and Tate's conjecture

We describe the behaviour of the rank of the Mordell-Weil group of the Picard variety of the generic fibre of a fibration in terms of local contributions given by averaging traces of Frobenius acting on the fibres. The results give a reinterpretation of Tate's conjecture (for divisors) and generalises previous results of Nagao, Rosen-Silverman and the authors.     

An answer to a conjecture on self-similar sets

In this paper, we construct a self-similar set which has a best covering but it is not the natural covering, thus negate the conjecture on self-similar sets posed by Z. Zhou in 2004.     

On the Cameron-Praeger conjecture

This paper takes a significant step towards confirming a long-standing and far-reaching conjecture of Peter J. Cameron and Cheryl E. Praeger. They conjectured in 1993 that there are no non-trivial block-transitive 6-designs. We prove that the Cameron-Praeger conjecture is true for the important case of non-trivial Steiner 6-designs, i.e. for 6-(v,k,λ) designs with λ=1, except possibly when the group is PΓL(2,pe) with p=2 or 3, and e is an odd prime power.     

混沌群作用

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Improvements on Hippchen's conjecture

Let G be a k-connected graph on n vertices. Hippchen's Conjecture (2008) states that two longest paths in G share at least k vertices. Gutiérrez (2020) recently proved the conjecture when $k\le 4$ or $k\ge \frac{n?2}{3}$. We improve upon both results; namely, we show that two longest paths in G share at least k vertices when $k=5$ or $k\ge \frac{n+2}{5}$. This completely resolves two conjectures by Gutiérrez in the affirmative.     

Frames and the Feichtinger conjecture

We show that the conjectured generalization of the Bourgain-Tzafriri restricted-invertibility theorem is equivalent to the conjecture of Feichtinger, stating that every bounded frame can be written as a finite union of Riesz basic sequences. We prove that any bounded frame can at least be written as a finite union of linearly independent sequences. We further show that the two conjectures are implied by the paving conjecture. Finally, we show that Weyl-Heisenberg frames over rational lattices are finite unions of Riesz basic sequences.

     

A note on the Jacobian conjecture

In this paper we consider the Jacobian conjecture for a map of complex affine spaces of dimension . It is well known that if is proper, then the conjecture will hold. Using topological arguments, specifically Smith theory, we show that the conjecture holds if and only if is proper onto its image.

     

Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero

This paper provides evidence for the Birch and Swinnerton-Dyer conjecture for analytic rank  abelian varieties  that are optimal quotients of attached to newforms. We prove theorems about the ratio , develop tools for computing with , and gather data about certain arithmetic invariants of the nearly abelian varieties of level . Over half of these have analytic rank , and for these we compute upper and lower bounds on the conjectural order of  . We find that there are at least such for which the Birch and Swinnerton-Dyer conjecture implies that is divisible by an odd prime, and we prove for of these that the odd part of the conjectural order of really divides by constructing nontrivial elements of using visibility theory. We also give other evidence for the conjecture. The appendix, by Cremona and Mazur, fills in some gaps in the theoretical discussion in their paper on visibility of Shafarevich-Tate groups of elliptic curves.