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Let M be a random rank-r matrix over the binary field , and let be its Hamming weight, that is, the number of nonzero entries of M.We prove that, as with r fixed and tending to a constant, we have that converges in distribution to a standard normal random variable. 相似文献
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In this paper, we establish a new asymptotic expansion of Gurland's ratio of gamma functions, that is, as ,where with and , , are the Bernoulli polynomials. Using a double inequality for hyperbolic functions, we prove that the function is completely monotonic on if , which yields a sharp upper bound for . This shows that the approximation for Gurland's ratio by the truncation of the above asymptotic expansion has a very high accuracy. We also present sharp lower and upper bounds for Gurland's ratio in terms of the partial sum of hypergeometric series. Moreover, some known results are contained in our results when . 相似文献
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《Discrete Mathematics》2021,344(12):112604
A well-known theorem of Vizing states that if G is a simple graph with maximum degree Δ, then the chromatic index of G is Δ or . A graph G is class 1 if , and class 2 if ; G is Δ-critical if it is connected, class 2 and for every . A long-standing conjecture of Vizing from 1968 states that every Δ-critical graph on n vertices has at least edges. We initiate the study of determining the minimum number of edges of class 1 graphs G, in addition, for every . Such graphs have intimate relation to -co-critical graphs, where a non-complete graph G is -co-critical if there exists a k-coloring of such that G does not contain a monochromatic copy of but every k-coloring of contains a monochromatic copy of for every . We use the bound on the size of the aforementioned class 1 graphs to study the minimum number of edges over all -co-critical graphs. We prove that if G is a -co-critical graph on vertices, then where ε is the remainder of when divided by 2. This bound is best possible for all and . 相似文献
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《Journal of Functional Analysis》2023,284(7):109835
We employ separation of variables to prove weighted resolvent estimates for the semiclassical Schrödinger operator in dimension , where , and is and compactly supported. The weighted resolvent norm grows no faster than , while an exterior weighted norm grows . We introduce a new method based on the Mellin transform to handle the two-dimensional case. 相似文献
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