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Uniform nonint egrability of random variables
Zechun HU, Xue PENG
Front. Math. China. 2018, 13 (1): 41-53.
https://doi.org/10.1007/s11464-017-0623-6
Recently, T. K. Chandra, T. -C. Hu and A. Rosalsky [Statist. Probab. Lett., 2016, 116: 27–37] introduced the notion of a sequence of random variables being uniformly nonintegrable, and presented a list of interesting results on this uniform nonintegrability. We introduce a weaker definition on uniform nonintegrability (W-UNI) of random variables, present a necessary and sufficient condition for W-UNI, and give two equivalent characterizations of WUNI, one of which is a W-UNI analogue of the celebrated de La Vallée Poussin criterion for uniform integrability. In addition, we give some remarks, one of which gives a negative answer to the open problem raised by Chandra et al.
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Super (a, d)-edge-antimagic total labelings of complete bipartite graphs
Zhihe LIANG
Front. Math. China. 2018, 13 (1): 129-146.
https://doi.org/10.1007/s11464-017-0671-y
An (a, d)-edge-antimagic total labeling of a graph G is a bijection f from V (G) ∪ E(G) onto {1, 2, . . . , |V (G)| +|E(G)|} with the property that the edge-weight set {f(x) + f(xy) + f(y) | xy ∈ E(G)} is equal to {a, a + d, a + 2d, . . . , a + (|E(G)| − 1)d} for two integers a>0 and d≥0. An (a, d)-edgeantimagic total labeling is called super if the smallest possible labels appear on the vertices. In this paper, we completely settle the problem of the super (a, d)-edge-antimagic total labeling of the complete bipartite graph Km,n and obtain the following results: the graph Km,n has a super (a, d)-edge-antimagic total labeling if and only if either (i) m = 1, n = 1, and d≥0, or (ii) m = 1, n≥2 (or n= 1 and m≥2), and d ∈ {0, 1, 2}, or (iii) m = 1, n = 2 (or n = 1 and m = 2), and d = 3, or (iv) m, n≥2, and d= 1.
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