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We study random graphs, both G( n,p) and G( n,m), with random orientations on the edges. For three fixed distinct vertices s,a,b we study the correlation, in the combine probability space, of the events $\{a\to s\}$ and $\{s\to b\}$ . For G(n,p), we prove that there is a $pc = 1/2$ such that for a fixed $p < pc$ the correlation is negative for large enough n and for $p > pc$ the correlation is positive for large enough n. We conjecture that for a fixed $n \ge 27$ the correlation changes sign three times for three critical values of p. For G(n,m) it is similarly proved that, with $p=m/({{n}\atop {2}})$ , there is a critical pc that is the solution to a certain equation and approximately equal to 0.7993. A lemma, which computes the probability of non existence of any $\ell$ directed edges in G(n,m), is thought to be of independent interest. We present exact recursions to compute \input amssym $\Bbb{P}(a\to s)$ and \input amssym $\Bbb{P}(a\to s, s\to b)$ . We also briefly discuss the corresponding question in the quenched version of the problem. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011 相似文献
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Partially supported by NSF Grant No. DMS-8803085 相似文献
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J. R. M. Mason 《Geometriae Dedicata》1984,15(4):355-361
A (k, d)-arc in PG(2, q) is a set of k points such that some d, but no d+1, of them are collinear. An outstanding problem is to find the maximum value of k for which a (k, d)-arc exists. A construction is given for a class of (k, p
n–p
m)-arcs in PG(2, p
n). These arcs constitute a lower bound on the maximum possible value of k, and a subset of them is shown to be optimal. 相似文献
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James Allen Fill Edward R. Scheinerman Karen B. Singer‐Cohen 《Random Structures and Algorithms》2000,16(2):156-176
When the random intersection graph G(n, m, p) proposed by Karoński, Scheinerman, and Singer‐Cohen [Combin Probab Comput 8 (1999), 131–159] is compared with the independent‐edge G(n, p), the evolutions are different under some values of m and equivalent under others. In particular, when m=nα and α>6, the total variation distance between the graph random variables has limit 0. ©2000 John Wiley & Sons, Inc. Random Struct. Alg., 16, 156–176, 2000 相似文献
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关于图B(m,n,p)的优美性 总被引:2,自引:1,他引:2
在[1][2]中已证明“除去三种特殊情形,连结两个顶点的三条独立路所成简单图B(m,n,p)是优美图”,并猜想:对除去的三种情形,B(m,n,p)也是优美的。本文证实了上术猜想。这样一来,也就证明了[3]中的猜想:有-h-链弦的圈是优美的(h≥2)。 相似文献
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Shen Guangyu 《数学年刊B辑(英文版)》1987,8(3):329-331
Supplementary discussions are given to the Lie algebra \Sigma(n, m, r, G). Minor errors in some formulas of a previous paper (see Chin. Ann. of Math., 4B(3), 1983, 329—346) are corrected. 相似文献
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O. I. Rudnitskii 《Journal of Mathematical Sciences》1996,82(2):3395-3398
We study the properties of the basis invariants of the symmetry groups of the complex polytope
and the generalized n-cube γ
n
m
, as well as its subgroups D
n
m
. We give an explicit construction of all the basis invariants of even degree of these groups.
Translated fromDinamicheskie Sistemy, Vol. 11, 1992. 相似文献
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D. Suryanarayana 《Aequationes Mathematicae》1978,18(1-2):322-329
In this paper, we discuss the pairs (f, h) of arithmetical functions satisfying the functional equation in the title, whereF is the product off andh under the Dirichlet convolution; that is,F(n) = Σ d|n ?(d)h(n/d) andS(m n) = Σd|(m, n) ?(d)h(n/d). The well-known Hölder's identity is a special case of this functional equation (?(n) =n, h(n) = μ(n)). We also generalize the functional equation in the title to any arbitrary regular arithmetical convolution and discuss the pairs of solutions (f, h) of the generalized functional equation and pose some problems relating to the characterization of all pairs of solutions. 相似文献
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关于(m,n,ρ)级半纯函数的一些结果 总被引:4,自引:0,他引:4
Theorem 1 If μ(r) is a continuous function for α≤r<∞, then there exist the function ρ(r) which satisfy the. following. 相似文献
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Brandt Kronholm 《The Ramanujan Journal》2013,30(3):425-436
Ramanujan-type congruences for the unrestricted partition function p(n) are well known and have been studied in great detail. The existence of Ramanujan-type congruences are virtually unknown for p(n,m), the closely related restricted partition function that enumerates the number of partitions of n into exactly m parts. Let ? be any odd prime. In this paper we establish explicit Ramanujan-type congruences for p(n,?) modulo any power of that prime ? α . In addition, we establish general congruence relations for p(n,?) modulo ? α for any n. 相似文献
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