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In this paper, we study the long-time behavior of solutions of a reaction–diffusion model in a one-dimensional river network, where the river network has two branches, and the water flow speeds in each branch are the same constant β. We show the existence of two critical values c0 and 2 with 0<c0<2, and prove that when c0β<2, the population density in every branch of the river goes to 1 as time goes to infinity; when 2<β<c0, then, as time goes to infinity, the population density in every river branch converges to a positive steady state strictly below 1; when |β|2, the species will be washed down the stream, and so locally the population density converges to 0. Our result indicates that only if the water-flow speed is suitably small (i.e., |β|<2), the species will survive in the long run.  相似文献   

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In Korchmáros et al. (2018)one-factorizations of the complete graph Kn are constructed for n=q+1 with any odd prime power q such that either q1(mod4) or q=2h?1. The arithmetic restriction n=q+1 is due to the fact that the vertices of Kn in the construction are the points of a conic Ω in the finite plane of order q. Here we work on the Euclidean plane and describe an analogous construction where the role of Ω is taken by a regular n-gon. This allows us to remove the above constraints and construct one-factorizations of Kn for every even n6.  相似文献   

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《Discrete Mathematics》2020,343(10):111996
A Gallai coloring of a complete graph Kn is an edge coloring without triangles colored with three different colors. A sequence e1ek of positive integers is an (n,k)-sequence if i=1kei=n2. An (n,k)-sequence is a G-sequence if there is a Gallai coloring of Kn with k colors such that there are ei edges of color i for all i,1ik. Gyárfás, Pálvölgyi, Patkós and Wales proved that for any integer k3 there exists an integer g(k) such that every (n,k)-sequence is a G-sequence if and only if ng(k). They showed that g(3)=5,g(4)=8 and 2k2g(k)8k2+1.We show that g(5)=10 and give almost matching lower and upper bounds for g(k) by showing that with suitable constants α,β>0, αk1.5lnkg(k)βk1.5 for all sufficiently large k.  相似文献   

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《Discrete Mathematics》2020,343(12):112127
Let r be a positive integer. The Bermond–Thomassen conjecture states that, a digraph of minimum out-degree at least 2r1 contains r vertex-disjoint directed cycles. A digraph D is called a local tournament if for every vertex x of D, both the out-neighbours and the in-neighbours of x 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 D with δ+(D)2r1 contains r disjoint cycles C1,C2,,Cr, satisfying that either Ci has the length at most 4 or is a shortest cycle of the original digraph of DC1Ci1 for 1ir.  相似文献   

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《Discrete Mathematics》2024,347(1):113657
A frequency n-cube Fn(q;l0,...,lm1) is an n-dimensional q-by-...-by-q array, where q=l0+...+lm1, filled by numbers 0,...,m1 with the property that each line contains exactly li cells with symbol i, i=0,...,m1 (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 m(q1)n. We improve that lower bound for n>2, replacing q1 by a smaller value s, by constructing a testing set of size sn 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   

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We further develop a forcing notion known as Coding with Perfect Trees and show that this poset preserves, in a strong sense, definable P-points, definable tight MAD families and definable selective independent families. As a result, we obtain a model in which a=u=i=?1<2?0=?2, each of a, u, i has a Π11 witness and there is a Δ31 well-order of the reals. Note that both the complexity of the witnesses of the above combinatorial cardinal characteristics, as well as the complexity of the well-order are optimal. In addition, we show that the existence of a Δ31 well-order of the reals is consistent with c=?2 and each of the following: a=u<i, a=i<u, a<u=i, where the smaller cardinal characteristics have co-analytic witnesses.Our methods allow the preservation of only sufficiently definable witnesses, which significantly differs from other preservation results of this type.  相似文献   

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Analogs of Waring–Hilbert problem on Cantor sets are explored. The focus of this paper is on the Cantor ternary set C. It is shown that, for each m3, every real number in the unit interval [0,1] is the sum x1m+x2m+?+xnm with each xj in C and some n6m. Furthermore, every real number x in the interval [0,8] can be written as x=x13+x23+?+x83, the sum of eight cubic powers with each xj in C. Another Cantor set C×C is also considered. More specifically, when C×C is embedded into the complex plane ?, the Waring–Hilbert problem on C×C has a positive answer for powers less than or equal to 4.  相似文献   

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A decomposition of a multigraph G is a partition of its edges into subgraphs G(1),,G(k). It is called an r-factorization if every G(i) is r-regular and spanning. If G is a subgraph of H, a decomposition of G is said to be enclosed in a decomposition of H if, for every 1ik, G(i) is a subgraph of H(i).Feghali and Johnson gave necessary and sufficient conditions for a given decomposition of λKn to be enclosed in some 2-edge-connected r-factorization of μKm for some range of values for the parameters n, m, λ, μ, r: r=2, μ>λ and either m2n?1, or m=2n?2 and μ=2 and λ=1, or n=3 and m=4. We generalize their result to every r2 and m2n?2. We also give some sufficient conditions for enclosing a given decomposition of λKn in some 2-edge-connected r-factorization of μKm for every r3 and m>(2?C)n, where C is a constant that depends only on r, λ and μ.  相似文献   

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