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1.
Qinhai ZHANG 《Frontiers of Mathematics in China》2022,17(1):1
We survey some unsolvable conjectures in finite p-groups and their research progress. 相似文献
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《Discrete Mathematics》2020,343(12):112127
Let be a positive integer. The Bermond–Thomassen conjecture states that, a digraph of minimum out-degree at least contains vertex-disjoint directed cycles. A digraph is called a local tournament if for every vertex of , both the out-neighbours and the in-neighbours of induce tournaments. Note that tournaments form the subclass of local tournaments. In this paper, we verify that the Bermond–Thomassen conjecture holds for local tournaments. In particular, we prove that every local tournament with contains disjoint cycles , satisfying that either has the length at most 4 or is a shortest cycle of the original digraph of for . 相似文献
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In 2001, J.-M. Le Bars disproved the zero-one law (that says that every sentence from a certain logic is either true asymptotically almost surely (a.a.s.), or false a.a.s.) for existential monadic second order sentences (EMSO) on undirected graphs. He proved that there exists an EMSO sentence ? such that does not converge as (here, the probability distribution is uniform over the set of all graphs on the labeled set of vertices ). In the same paper, he conjectured that, for EMSO sentences with 2 first order variables, the zero-one law holds. In this paper, we disprove this conjecture. 相似文献
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Let k be an algebraically closed field and A the polynomial algebra in r variables with coefficients in k. In case the characteristic of k is 2, Carlsson [9] conjectured that for any DG-A-module M of dimension N as a free A-module, if the homology of M is nontrivial and finite dimensional as a k-vector space, then . Here we state a stronger conjecture about varieties of square-zero upper triangular matrices with entries in A. Using stratifications of these varieties via Borel orbits, we show that the stronger conjecture holds when or without any restriction on the characteristic of k. As a consequence, we obtain a new proof for many of the known cases of Carlsson's conjecture and give new results when and . 相似文献
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In this note, it is shown that the validity of the Auslander–Reiten conjecture for a given d-dimensional Cohen–Macaulay local ring R depends on its validity for all direct summands of d-th syzygy of R-modules of finite length, provided R is an isolated singularity. Based on this result, it is shown that under a mild assumption on the base ring R, satisfying the Auslander–Reiten conjecture behaves well under completion and reduction modulo regular elements. In addition, it will turn out that, if R is a commutative Noetherian ring and 𝒬 a finite acyclic quiver, then the Auslander–Reiten conjecture holds true for the path algebra R𝒬, whenever so does R. Using this result, examples of algebras satisfying the Auslander–Reiten conjecture are presented. 相似文献