共查询到20条相似文献,搜索用时 15 毫秒
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《Journal of Functional Analysis》2023,284(9):109877
We prove an atomic type decomposition for the noncommutative martingale Hardy space for all by an explicit constructive method using algebraic atoms as building blocks. Using this elementary construction, we obtain a weak form of the atomic decomposition of for all , and provide a constructive proof of the atomic decomposition for which resolves a main problem on the subject left open for the last twelve years. We also study -atoms, and show that every -atom can be decomposed into a sum of -atoms; consequently, for every , the -atoms lead to the same atomic space for all . As applications, we obtain a characterization of the dual space of the noncommutative martingale Hardy space () as a noncommutative Lipschitz space via the weak form of the atomic decomposition. Our constructive method can also be applied to prove some sharp martingale inequalities. 相似文献
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《Discrete Mathematics》2022,345(9):112977
Consider functions , where A and C are disjoint finite sets. The weakly connected components of the digraph of such a function are cycles of rooted trees, as in random mappings, and isolated rooted trees. Let and . When a function is chosen from all possibilities uniformly at random, then we find the following limiting behaviour as . If , then the size of the maximal mapping component goes to infinity almost surely; if , a constant, then process counting numbers of mapping components of different sizes converges; if , then the number of mapping components converges to 0 in probability. We get estimates on the size of the largest tree component which are of order when and constant when , . These results are similar to ones obtained previously for random injections, for which the weakly connected components are cycles and linear trees. 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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《Discrete Mathematics》2023,346(4):113304
In 1965 Erd?s asked, what is the largest size of a family of k-element subsets of an n-element set that does not contain a matching of size ? In this note, we improve upon a recent result of Frankl and resolve this problem for and . 相似文献
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《Discrete Mathematics》2024,347(1):113657
A frequency n-cube is an n-dimensional q-by-...-by-q array, where , filled by numbers with the property that each line contains exactly cells with symbol i, (a line consists of q cells of the array differing in one coordinate). The trivial upper bound on the number of frequency n-cubes is . We improve that lower bound for , replacing by a smaller value s, by constructing a testing set of size for frequency n-cubes (a testing set is a collection of cells of an array the values in which uniquely determine the array with given parameters). We also construct new testing sets for generalized frequency n-cubes, which are essentially correlation-immune functions in n q-valued arguments; the cardinalities of new testing sets are smaller than for testing sets known before. 相似文献