共查询到20条相似文献,搜索用时 31 毫秒
1.
Andreas Weingartner 《Journal of Number Theory》2004,108(1):18-28
Let 1=d1(n)<d2(n)<?<dτ(n)=n be the sequence of all positive divisors of the integer n in increasing order. We say that the divisors of n are t-dense iff max1?i<τ(n)di+1(n)/di(n)?t. Let D(x,t) be the number of positive integers not exceeding x whose divisors are t-dense. We show that for x?3, and , we have , where , and d(w) is a continuous function which satisfies d(w)?1/w for w?1. We also consider other counting functions closely related to D(x,t). 相似文献
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Tom Halverson 《Journal of Combinatorial Theory, Series A》2010,117(5):507-72
We study a q-analog Qr(n,q) of the partition algebra Pr(n). The algebra Qr(n,q) arises as the centralizer algebra of the finite general linear group GLn(Fq) acting on a vector space coming from r-iterations of Harish-Chandra restriction and induction. For n?2r, we show that Qr(n,q) has the same semisimple matrix structure as Pr(n). We compute the dimension to be a q-polynomial that specializes as dn,r(1)=nr and dn,r(0)=B(r), the rth Bell number. Our method is to write dn,r(q) as a sum over integer sequences which are q-weighted by inverse major index. We then find a basis of indexed by n-restricted q-set partitions of {1,…,r} and show that there are dn,r(q) of these. 相似文献
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Andreas Weingartner 《Journal of Number Theory》2004,108(1):1-17
Let 1=d1(n)<d2(n)<?<dτ(n)=n be the sequence of all positive divisors of the integer n in increasing order. We say that the divisors of n are y-dense iff max1?i<τ(n)di+1(n)/di(n)?y. Let D(x,y,z) be the number of positive integers not exceeding x whose divisors are y-dense and whose prime divisors are bigger than z, and let , and . We show that is equivalent, in a large region, to a function d(u,v) which satisfies a difference-differential equation. Using that equation we find that d(u,v)?(1−u/v)/(u+1) for v?3+ε. Finally, we show that d(u,v)=e−γd(u)+O(1/v), where γ is Euler's constant and d(u)∼x−1D(x,y,1), for fixed u. This leads to a new estimate for d(u). 相似文献
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Yihong Song Christopher T.H. Baker 《Journal of Mathematical Analysis and Applications》2004,294(1):310-333
Various different types of stability are defined, in a unified framework, for discrete Volterra equations of the type x(n)=f(n)+∑nj=0K(n,j,x(n)) (n?0). Under appropriate assumptions, stability results are obtainable from those valid in the linear case (K(n,j,x(n))=B(n,j)x(j)), and a linearized stability theory is studied here by using the fundamental and resolvent matrices. Several necessary and sufficient conditions for stability are obtained for solutions of the linear equation by considering the equations in various choices of Banach space , the elements of which are sequences of vectors (, , n,j?0, etc.). We show that the theory, including a number of new results as well as results already known, can be presented in a systematic framework, in which results parallel corresponding results for classical Volterra integral equations of the second kind. 相似文献
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Let d(σ) stand for the defining number of the colouring σ. In this paper we consider and for the onto χ-colourings γ of the circular complete graph Kn,d. In this regard we obtain a lower bound for dmin(Kn,d) and we also prove that this parameter is asymptotically equal to χ-1. Also, we show that when χ?4 and s≠0 then dmax(Kχd-s,d)=χ+2s-3, and, moreover, we prove an inequality relating this parameter to the circular chromatic number for any graph G. 相似文献
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A nonincreasing sequence of nonnegative integers π=(d1,d2,…,dn) is graphic if there is a (simple) graph G of order n having degree sequence π. In this case, G is said to realizeπ. For a given graph H, a graphic sequence π is potentiallyH-graphic if there is some realization of π containing H as a (weak) subgraph. Let σ(π) denote the sum of the terms of π. For a graph H and n∈Z+, σ(H,n) is defined as the smallest even integer m so that every n-term graphic sequence π with σ(π)≥m is potentially H-graphic. Let denote the complete t partite graph such that each partite set has exactly s vertices. We show that and obtain the exact value of σ(Kj+Ks,s,n) for n sufficiently large. Consequently, we obtain the exact value of for n sufficiently large. 相似文献
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A polygon is an elementary (self-avoiding) cycle in the hypercubic lattice dtaking at least one step in every dimension. A polygon on dis said to be convex if its length is exactly twice the sum of the side lengths of the smallest hypercube containing it. The number ofd-dimensional convex polygonspn, dof length 2nwithd(n)→∞ is asymptoticallywherer=r(n, d) is the unique solution ofr coth r=2n/d−1andb(r)=d(r coth r−r2/sinh2 r). The convergence is uniform over alld?ω(n) for any functionω(n)→∞. Whendis constant the exponential is replaced with (1−d−1)2d. These results are proved by asymptotically enumerating a larger class of combinatorial objects calledconvex proto-polygonsby the saddle-point method and then finding the asymptotic probability a randomly chosen convex proto-polygon is a convex polygon. 相似文献
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Shaun Cooper 《Journal of Number Theory》2003,103(2):135-162
Let rk(n) denote the number of representations of an integer n as a sum of k squares. We prove that for odd primes p,
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Zhi-Hong Sun 《Journal of Number Theory》2005,114(1):88-123
If and are two sequences such that a1=b1 and , then we say that (an,bn) is a Newton-Euler pair. In the paper, we establish many formulas for Newton-Euler pairs, and then make use of them to obtain new results concerning some special sequences such as and Bn, where p(n) is the number of partitions of n, σ(n) is the sum of divisors of n, and Bn is the nth Bernoulli number. 相似文献
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Paul Pollack 《Journal of Number Theory》2010,130(8):1732-1736
Write s(n) for the sum of the proper divisors of the natural number n. We call n sociable if the sequence n, s(n), s(s(n)), … is purely periodic; the period is then called the order of sociability of n. The ancients initiated the study of order 1 sociables (perfect numbers) and order 2 sociables (amicable numbers), and investigations into higher-order sociable numbers began at the end of the 19th century. We show that if k is odd and fixed, then the number of sociable n?x of order k is bounded by as x→∞. This improves on the previously best-known bound of , due to Kobayashi, Pollack, and Pomerance. 相似文献
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Given a finite set of 2-dimensional points P⊆R2 and a positive real d, a unit disk graph, denoted by (P,d), is an undirected graph with vertex set P such that two vertices are adjacent if and only if the Euclidean distance between the pair is less than or equal to d. Given a pair of non-negative integers m and n, P(m,n) denotes a subset of 2-dimensional triangular lattice points defined by where . Let Tm,n(d) be a unit disk graph defined on a vertex set P(m,n) and a positive real d. Let be the kth power of Tm,n(1).In this paper, we show necessary and sufficient conditions that [ is perfect] and/or [ is perfect], respectively. These conditions imply polynomial time approximation algorithms for multicoloring (Tm,n(d),w) and . 相似文献
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Akihiko Inoue 《Journal of multivariate analysis》2004,89(1):135-147
Let be a fractional ARIMA(p,d,q) process with partial autocorrelation function α(·). In this paper, we prove that if d∈(−1/2,0) then |α(n)|∼|d|/n as n→∞. This extends the previous result for the case 0<d<1/2. 相似文献
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Jian-Hua Yin 《Discrete Mathematics》2009,309(8):2579-6276
An r-graph is a loopless undirected graph in which no two vertices are joined by more than r edges. An r-complete graph on m+1 vertices, denoted by , is an r-graph on m+1 vertices in which each pair of vertices is joined by exactly r edges. A non-increasing sequence π=(d1,d2,…,dn) of nonnegative integers is r-graphic if it is realizable by an r-graph on n vertices. Let be the smallest even integer such that each n-term r-graphic sequence with term sum of at least is realizable by an r-graph containing as a subgraph. In this paper, we determine the value of for sufficiently large n, which generalizes a conjecture due to Erd?s, Jacobson and Lehel. 相似文献
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The fractional weak discrepancywdF(P) of a poset P=(V,?) was introduced in [A. Shuchat, R. Shull, A. Trenk, The fractional weak discrepancy of a partially ordered set, Discrete Applied Mathematics 155 (2007) 2227-2235] as the minimum nonnegative k for which there exists a function satisfying (i) if a?b then f(a)+1≤f(b) and (ii) if a∥b then |f(a)−f(b)|≤k. In this paper we generalize results in [A. Shuchat, R. Shull, A. Trenk, Range of the fractional weak discrepancy function, ORDER 23 (2006) 51-63; A. Shuchat, R. Shull, A. Trenk, Fractional weak discrepancy of posets and certain forbidden configurations, in: S.J. Brams, W.V. Gehrlein, F.S. Roberts (Eds.), The Mathematics of Preference, Choice, and Order: Essays in Honor of Peter C. Fishburn, Springer, New York, 2009, pp. 291-302] on the range of the wdF function for semiorders (interval orders with no induced ) to interval orders with no , where n≥3. In particular, we prove that the range for such posets P is the set of rationals that can be written as r/s, where 0≤s−1≤r<(n−2)s. If wdF(P)=r/s and P has an optimal forcing cycle C with and , then r≤(n−2)(s−1). Moreover when s≥2, for each r satisfying s−1≤r≤(n−2)(s−1) there is an interval order having such an optimal forcing cycle and containing no. 相似文献
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Let G=(V,E) be a simple graph with vertex degrees d1,d2,…,dn. The Randi? index R(G) is equal to the sum over all edges (i,j)∈E of weights . We prove several conjectures, obtained by the system AutoGraphiX, relating R(G) and the chromatic number χ(G). The main result is χ(G)≤2R(G). To prove it, we also show that if v∈V is a vertex of minimum degree δ of G, G−v the graph obtained from G by deleting v and all incident edges, and Δ the maximum degree of G, then . 相似文献
20.
Rafa? Filipów 《Journal of Mathematical Analysis and Applications》2010,362(1):64-71
We consider ideals I of subsets of the set of natural numbers such that for every conditionally convergent series ∑n∈ωan and every there is a permutation such that ∑n∈ωaπr(n)=r and