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1.
Jian Wang 《高校应用数学学报(英文版)》2008,23(3):345-350
A K1,k-factorization of λKm,n is a set of edge-disjoint K1,k-factors of λKm,n, which partition the set of edges of λKm,n. In this paper, it is proved that a sufficient condition for the existence of K1,k-factorization of λKm,n, whenever k is any positive integer, is that (1) m ≤ kn, (2) n ≤ km, (3) km-n = kn-m ≡ 0 (mod (k^2- 1)) and (4) λ(km-n)(kn-m) ≡ 0 (mod k(k- 1)(k^2 - 1)(m + n)). 相似文献
2.
DuBeiliang 《高校应用数学学报(英文版)》2001,16(2):107-110
Abstract. In this paper, it is shown that a sufficient condition for the existence of a 相似文献
3.
Intersection theorems with geometric consequences 总被引:3,自引:0,他引:3
In this paper we prove that ifℱ is a family ofk-subsets of ann-set, μ0, μ1, ..., μs are distinct residues modp (p is a prime) such thatk ≡ μ0 (modp) and forF ≠ F′ ≠ℱ we have |F ∩F′| ≡ μi (modp) for somei, 1 ≦i≦s, then |ℱ|≦(
s
n
).
As a consequence we show that ifR
n
is covered bym sets withm<(1+o(1)) (1.2)
n
then there is one set within which all the distances are realised.
It is left open whether the same conclusion holds for compositep. 相似文献
4.
Let (v,u×c,λ)-splitting BIBD denote a (v,u×c,λ)-splitting balanced incomplete block design of order v with block size u×c and index λ. Necessary conditions for the existence of a (v,u×c,λ)-splitting BIBD are v≥uc, λ(v−1)≡0 (mod c(u−1)) and λ
v(v−1)≡0 (mod (c
2
u(u−1))). We show in this paper that the necessary conditions for the existence of a (v,3×3,λ)-splitting BIBD are also sufficient with possible exceptions when (1) (v,λ)∈{(55,1),(39,9k):k=1,2,…}, (2) λ≡0 (mod 54) and v≡0 (mod 2). We also show that there exists a (v,3×4,1)-splitting BIBD when v≡1 (mod 96). As its application, we obtain a new infinite class of optimal 4-splitting authentication codes. 相似文献
5.
In this paper, it is shown that a necessary and sufficient condition for the existence of aP
3-factorization ofK
m
n
is (i)mn 0(mod 3) and (ii) (m – 1)n 0(mod 4). 相似文献
6.
It is shown that ifA andB are non-empty subsets of {0, 1}
n
(for somenεN) then |A+B|≧(|A||B|)α where α=(1/2) log2 3 here and in what follows. In particular if |A|=2
n-1 then |A+A|≧3
n-1 which anwers a question of Brown and Moran. It is also shown that if |A| = 2
n-1 then |A+A|=3
n-1 if and only if the points ofA lie on a hyperplane inn-dimensions. Necessary and sufficient conditions are also given for |A +B|=(|A||B|)α. The above results imply the following improvement of a result of Talagrand [7]: ifX andY are compact subsets ofK (the Cantor set) withm(X),m(Y)>0 then λ(X+Y)≧2(m(X)m(Y))α wherem is the usual measure onK and λ is Lebesgue measure. This also answers a question of Moran (in more precise terms) showing thatm is not concentrated on any proper Raikov system. 相似文献
7.
Let λK
m,n be a bipartite multigraph with two partite sets having m and n vertices, respectively. A P
v-factorization of λK
m,n is a set of edge-disjoint P
v
-factors of λK
m,n which partition the set of edges of λK
m,n. When v is an even number, Ushio, Wang and the second author of the paper gave a necessary and sufficient condition for the existence
of a P
v
-factorization of λK
m,n. When v is an odd number, we proposed a conjecture. However, up to now we only know that the conjecture is true for v = 3. In this paper we will show that the conjecture is true when v = 4k − 1. That is, we shall prove that a necessary and sufficient condition for the existence of a P
4k−1-factorization of λK
m,n is (1) (2k − 1)m ⩽ 2kn, (2) (2k − 1)n ⩽ 2km, (3) m + n ≡ 0 (mod 4k − 1), (4) λ(4k − 1)mn/[2(2k − 1)(m + n)] is an integer. 相似文献
8.
Shen Hao 《数学学报(英文版)》1993,9(3):246-251
It is proved in this paper that there exists an incomplete Mendelsohn triple system IMTS(u,v; λ) if and only ifλ(u-v)(u-2v-1)≡0(mod 3),u≥2v+1 and (u, v, λ) ≠ (6, 1, 1). As a consequence, it is proved that for any given λ≥1, a Mendelsohn triple system MTS (v, λ) can be embedded in an MTS (u, λ) if and only ifλu(u-1)≡0(mod 3) andu≥2v+1.
Project supported by the National Natural Science Foundation of China. 相似文献
9.
The spectrum of path factorization of bipartite multigraphs 总被引:1,自引:0,他引:1
Jian WANG~ Bei-liang DU~ 《中国科学A辑(英文版)》2007,50(7):1045-1054
LetλK_(m,n)be a bipartite multigraph with two partite sets having m and n vertices, respectively.A P_v-factorization ofλK_(m,n)is a set of edge-disjoint P_v-factors ofλK_(m,n)which partition the set of edges ofλK_(m,n).When v is an even number,Ushio,Wang and the second author of the paper gave a necessary and sufficient condition for the existence of a P_v-factorization ofλK_(m,n).When v is an odd number,we have proposed a conjecture.Very recently,we have proved that the conjecture is true when v=4k-1.In this paper we shall show that the conjecture is true when v = 4k 1,and then the conjecture is true.That is,we will prove that the necessary and sufficient conditions for the existence of a P_(4k 1)-factorization ofλK_(m,n)are(1)2km≤(2k 1)n,(2)2kn≤(2k 1)m,(3)m n≡0(mod 4k 1),(4)λ(4k 1)mn/[4k(m n)]is an integer. 相似文献
10.
In this paper we obtain some new identities containing Fibonacci and Lucas numbers. These identities allow us to give some
congruences concerning Fibonacci and Lucas numbers such as L
2mn+k
≡ (−1)(m+1)n
L
k
(mod L
m
), F
2mn+k
≡ (−1)(m+1)n
F
k
(mod L
m
), L
2mn+k
≡ (−1)
mn
L
k
(mod F
m
) and F
2mn+k
≡ (−1)
mn
F
k
(mod F
m
). By the achieved identities, divisibility properties of Fibonacci and Lucas numbers are given. Then it is proved that there
is no Lucas number L
n
such that L
n
= L
2
k
t
L
m
x
2 for m > 1 and k ≥ 1. Moreover it is proved that L
n
= L
m
L
r
is impossible if m and r are positive integers greater than 1. Also, a conjecture concerning with the subject is given. 相似文献
11.
12.
Gennian Ge Malcolm Greig Jennifer Seberry Ralph Seberry 《Graphs and Combinatorics》2007,23(3):271-290
We show that if G is a finite Abelian group and the block size is 3, then the necessary conditions for the existence of a (v,3,λ;G) GBRD are sufficient. These necessary conditions include the usual necessary conditions for the existence of the associated
(v,3,λ) BIBD plus λ≡ 0 (mod|G|), plus some extra conditions when |G| is even, namely that the number of blocks be divisible by 4 and, if v = 3 and the Sylow 2-subgroup of G is cyclic, then also λ≡ 0 (mod2|G|). 相似文献
13.
Hei-Chi Chan 《数学学报(英文版)》2011,27(4):625-634
In this paper, we study a certain partition function a(n) defined by Σ
n≥0
a(n)q
n
:= Π
n=1(1 − q
n
)−1(1 − q
2n
)−1. We prove that given a positive integer j ≥ 1 and a prime m ≥ 5, there are infinitely many congruences of the type a(An + B) ≡ 0 (mod m
j
). This work is inspired by Ono’s ground breaking result in the study of the distribution of the partition function p(n). 相似文献
14.
We obtain the span of the real flag manifolds ℝF(1, 1, n−2), n ≥ 3, for the cases n ≡ 2 (mod 4), n ≡ 4 (mod 8) and n ≡ 8 (mod 16) and use the results to deduce that certain Stiefel-Whitney classes of the manifold are zero.
相似文献
15.
An ordered analogue of quadruple systems is tetrahedral quadruple systems. A tetrahedral quadruple system of order v and index λ, TQS(v, λ), is a pair (S, T){(S, \mathcal{T})} where S is a finite set of v elements and T{\mathcal{T}} is a family of oriented tetrahedrons of elements of S called blocks, such that every directed 3-cycle on S is contained in exactly λ blocks of T{\mathcal{T}} . When λ = 1, the spectrum problem of TQS(v, 1) has been completely determined. It is proved that a TQS(v, λ) exists if and only if λ(v − 1)(v − 2) ≡ 0 (mod 3), λv(v − 1)(v − 2) ≡ 0 (mod 4) and v ≥ 4. 相似文献
16.
Massimo Giulietti 《Designs, Codes and Cryptography》2008,47(1-3):135-143
For any divisor k of q
4−1, the elements of a group of k
th-roots of unity can be viewed as a cyclic point set C
k
in PG(4,q). An interesting problem, connected to the theory of BCH codes, is to determine the spectrum A(q) of maximal divisors k of q
4−1 for which C
k
is a cap. Recently, Bierbrauer and Edel [Edel and Bierbrauer (2004) Finite Fields Appl 10:168–182] have proved that 3(q
2 + 1)∈A(q) provided that q is an even non-square. In this paper, the odd order case is investigated. It is proved that the only integer m for which m(q
2 + 1)∈A(q) is m = 2 for q ≡ 3 (mod 4), m = 1 for q ≡ 1 (mod 4). It is also shown that when q ≡ 3 (mod 4), the cap is complete.
相似文献
17.
Extending the analogous result of Cannon and Wagreich for the fundamental groups of surfaces, we show that, for the -regular graphs
associated to regular tessellations of the hyperbolic plane by m-gons, the denominators of the growth series (which are rational and were computed by Floyd and Plotnick) are reciprocal Salem polynomials. As a consequence, the growth rates of these graphs are Salem numbers. We also prove that these denominators are essentially irreducible (they have a factor of X + 1 when m 2 mod 4; and when = 3 and m 4 mod 12, for instance, they have a factor of X
2 – X + 1). We then derive some regularity properties for the coefficients f
n
of the growth series: they satisfy K
n
– R < f
n
< K
n
+ R for some constants K, R < 0, < 1. 相似文献
18.
J. D. Fanning 《Aequationes Mathematicae》1994,47(2-3):143-149
Summary We prove the following two non-existence theorems for symmetric balanced ternary designs. If 1 = 1 and 0 (mod 4) then eitherV = + 1 or 42 – + 1 is a square and (42 – + 1) divides 2 – 1. If 1 = 2 thenV = ((m + 1)/2)
2 + 2,K = (m
2 + 7)/4 and = ((m – 1)/2)2 + 1 wherem 3 (mod 4). An example belonging to the latter series withV = 18 is constructed. 相似文献
19.
Li-dong Wang Hai-rong Kong Hong-juan Liu Department of Basic Courses Chinese People’s Armed Police Force Academy Langfang China School of Science Hebei University of Technology Tianjin China Department of Computer Science Engineering Langfang Polytechnic Institute China 《应用数学学报(英文版)》2011,27(3):407-418
In this paper, we investigate the existence of incomplete group divisible designs (IGDDs) with block size four, group-type (g, h) u and general index λ. The necessary conditions for the existence of such a design are that u ≥ 4, g ≥ 3h, λg(u 1) ≡ 0 (mod 3), λ(g h)(u 1) ≡ 0 (mod 3), and λu(u 1)(g 2 h 2 ) ≡ 0 (mod 12). These necessary conditions are shown to be sufficient for all λ≥ 2. The known existence result for λ = 1 is also improved. 相似文献
20.
The basic necessary conditions for the existence of a (v, k, λ)-perfect Mendelsohn design (briefly (v, k, λ)-PMD) are v ≥ k and λ v(v − 1) ≡ 0 (mod k). These conditions are known to be sufficient in most cases, but certainly not in all. For k = 3, 4, 5, 7, very extensive investigations of (v, k, λ)-PMDs have resulted in some fairly conclusive results. However, for k = 6 the results have been far from conclusive, especially for the case of λ = 1, which was given some attention in papers
by Miao and Zhu [34], and subsequently by Abel et al. [1]. Here we investigate the situation for k = 6 and λ > 1. We find that the necessary conditions, namely v ≥ 6 and λ v(v − 1)≡0 (mod 6) are sufficient except for the known impossible cases v = 6 and either λ = 2 or λ odd.
Researcher F.E. Bennett supported by NSERC Grant OGP 0005320. 相似文献