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Let be a rotation on the circle and let φ be a step function. Denote by the ergodic sums . For α in a class containing the rotations with bounded partial quotients and under a Diophantine condition on the discontinuities of φ, we show that is asymptotically Gaussian for n in a set of density 1. 相似文献
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《Discrete Mathematics》2022,345(10):113004
Let G be a graph. We say that G is perfectly divisible if for each induced subgraph H of G, can be partitioned into A and B such that is perfect and . We use and to denote a path and a cycle on t vertices, respectively. For two disjoint graphs and , we use to denote the graph with vertex set and edge set , and use to denote the graph with vertex set and edge set . In this paper, we prove that (i) -free graphs are perfectly divisible, (ii) if G is -free with , (iii) if G is -free, and (iv) if G is -free. 相似文献
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In this paper we provide the sharp conditions of the uniqueness for inverse nodal Sturm–Liouville problems defined on interval with separated boundary conditions. We prove that the potential and boundary parameters can be uniquely determined by a dense nodal subset contained on with through two cases of and , where in the latter case the nodal subset also need to be paired. Note that, the dense nodal subset was required to be twin for both cases in the previous works. 相似文献
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《Discrete Mathematics》2022,345(8):112903
Graphs considered in this paper are finite, undirected and loopless, but we allow multiple edges. The point partition number is the least integer k for which G admits a coloring with k colors such that each color class induces a -degenerate subgraph of G. So is the chromatic number and is the point arboricity. The point partition number with was introduced by Lick and White. A graph G is called -critical if every proper subgraph H of G satisfies . In this paper we prove that if G is a -critical graph whose order satisfies , then G can be obtained from two non-empty disjoint subgraphs and by adding t edges between any pair of vertices with and . Based on this result we establish the minimum number of edges possible in a -critical graph G of order n and with , provided that and t is even. For the corresponding two results were obtained in 1963 by Tibor Gallai. 相似文献