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1.
Let D(v) dente the maximum number of quintuples of a v-set of points X with the property that every pair of distinct points of X occurs in at most one quintuple,Let B(v)=[v(v-1)/4]/5],It is shown is this paper that D(v)=B(v) for all v≡0 (mod 4)with 2 exceptions and 13 possible exceptions.  相似文献   

2.
Let p ≡ 2(mod 3) be an odd prime and α be a positive integer. In this paper,for any integer c, we obtain a formula for the number of solutions of the cubic congruence x~3+ y~3≡ c(mod p~α) with x, y units, nonunits and mixed pairs, respectively. We resolve a problem posed by Yang and Tang.  相似文献   

3.
If q is an odd integer, q≥3,for any integers α, (α,q) = 1,there exsits a positiveinteger α, so tbat αα≡1(mod q) and 1≤α≤q - 1. Let L(q) = {α|α∈Z,1≤α≤q - 1, (α,q) = 1 and α +α=1(mod 2)}. (1)About the property of elements of L(q) is a generalization of a problem of D. H.Lenmer ([1],p. 12). In [3], it was conjectured that  相似文献   

4.
Let A(n) be the largest absolute value of any coefficient of n-th cyclotomic polynomial Φn(x).We say Φn(x) is flat if A(n) = 1.In this paper,for odd primes p q r and 2r ≡ 1(mod pq),we prove that Φpqr(x) is flat if and only if p = 3 and q ≡ 1(mod 3).  相似文献   

5.
In this paper,we investigate the intersection numbers of nearly Kirkman triple systems.J_N [v] is the set of all integers k such that there is a pair of NKTS(v)s with a common uncovered collection of 2-subset intersecting in k triples.It has been established that J_N[v]={0,1,...,v(v-2)/6-6,v(v-2)/6-4,v(v-2)/6} for any integers v ≡ 0(mod 6) and v≥66.For v≤60,there are 8 cases left undecided.  相似文献   

6.
Given any set K of positive integers and positive integer λ, let c(K,λ) denote the smallest integer such that v∈B(K,λ) for every integer v≥c(K,λ) that satisfies the congruences λv(v-1)≡0 (mod β(K) and λ(v-1)≡0 (mod α(K)). Let K0 be an equivalent set of K, k and k* be the smallest and the largest integers in K0. We prove that c(K,λ)≤exp exp{Q0}Qo=max{2(2p(ko)2-k2kk)p(ko)4,(Kk242y-k-2)(y2)}, whereand y=k*+k(k-1)+1.  相似文献   

7.
Let p be an odd prime and let a,m ∈ Z with a 0 and p ︱ m.In this paper we determinep ∑k=0 pa-1(2k k=d)/mk mod p2 for d=0,1;for example,where(-) is the Jacobi symbol and {un}n≥0 is the Lucas sequence given by u0 = 0,u1 = 1 and un+1 =(m-2)un-un-1(n = 1,2,3,...).As an application,we determine ∑0kpa,k≡r(mod p-1) Ck modulo p2 for any integer r,where Ck denotes the Catalan number 2kk /(k + 1).We also pose some related conjectures.  相似文献   

8.
For the Diophantine equation x^4 — Dy^2 = 1 (1) where D>0 and is not a perfect square, we prove the following theorems in this paper. Theorem 1. If D\[{\not \equiv }\]7 (mod 8),D=p1p2...ps,s≥2,where pi(i = 1,…,s) are distincyt primes,p1≡1(mod 4) such that either 2p1=a^2+b^2,а≡\[ \pm \]3(mod 8),b三\[ \pm \]3(mod 8) or there is a j(2≤j≤s), for which Legendre symbal \[\left( {\frac{{{p_j}}}{{{p_1}}}} \right) = - 1\],and pi≡7(mod8) (i=2,..., s) or pi≡3(mod 8) (i=2,..., s), then (1) has no solutions in positive integer x,y. Theorem 2. If D=p1...ps,s≥2, where pi(i = 1,…,s) are distinct primes, and pi≡3(mod 4)(i = 1,…,s), then (1) has no solutions in positive integer x, y. Theorem 3. The equation (1) with D=2p1...ps has no solutions in positive integer x, y, if (1) p1≡(mod 4), pi≡7(mod 8) (i = 2, ???, s), snch that either 2p1 = a^2+b^2 a≡\[ \pm \]3(mod 8),b≡\[ \pm \]3(mod 8)or there is a j (2≤j≤s),for which \[\left( {\frac{{{p_j}}}{{{p_1}}}} \right) = - 1\]; or (2) p1≡5(mod8),pi≡3(mod8) (i = 2,..., s); or ⑶p1≡5(mod8),pi≡7(mod 8) (i=2,…,s). Corollary of theorem 3. If D = 2pq, p≡5(mod 8), q≡3(mod 4), where p, q are distinct primes, then (1) has no solutions in positive integer x, y. Theorem 4. If D=2p1...ps, pi≡3(mod 4)(0 = 1,...,s), then (1) has no solutions In positive integer x, y.  相似文献   

9.
A family (X, B1), (X, B2), . . . , (X, Bq) of q STS(v)s is a λ-fold large set of STS(v) and denoted by LSTSλ(v) if every 3-subset of X is contained in exactly λ STS(v)s of the collection. It is indecomposable and denoted by IDLSTSλ(v) if there exists no LSTSλ (v) contained in the collection for any λ λ. In 1995, Griggs and Rosa posed a problem: For which values of λ 1 and orders v ≡ 1, 3 (mod 6) do there exist IDLSTSλ(v)? In this paper, we use partitionable candelabra systems (PCSs) and holey λ-fold large set of STS(v) (HLSTSλ(v)) as auxiliary designs to establish a recursive construction for IDLSTSλ(v) and show that there exists an IDLSTSλ(v) for λ = 2, 3, 4 and v ≡ 1, 3 (mod 6).  相似文献   

10.
Let(a, b, c) be a primitive Pythagorean triple. Je′smanowicz conjectured in 1956 that for any positive integer n, the Diophantine equation(an)x+(bn)y=(cn)z has only the positive integer solution(x, y, z) =(2, 2, 2). Let p ≡ 3(mod 4) be a prime and s be some positive integer. In the paper, we show that the conjecture is true when(a, b, c) =(4p2s-1, 4p s, 4p2s+ 1) and certain divisibility conditions are satisfied.  相似文献   

11.
For bipartite graphs G 1, G 2, . . . ,G k , the bipartite Ramsey number b(G 1, G 2, . . . , G k ) is the least positive integer b so that any colouring of the edges of K b,b with k colours will result in a copy of G i in the ith colour for some i. A tree of diameter three is called a bistar, and will be denoted by B(s, t), where s ≥ 2 and t ≥ 2 are the degrees of the two support vertices. In this paper we will obtain some exact values for b(B(s, t), B(s, t)) and b(B(s, s), B(s, s)). Furtermore, we will show that if k colours are used, with k ≥ 2 and s ≥ 2, then \({b_{k}(B(s, s)) \leq \lceil k(s - 1) + \sqrt{(s - 1)^{2}(k^{2} - k) - k(2s - 4)} \rceil}\) . Finally, we show that for s ≥ 3 and k ≥ 2, the Ramsey number \({r_{k}(B(s, s)) \leq \lceil 2k(s - 1)+ \frac{1}{2} + \frac{1}{2} \sqrt{(4k(s - 1) + 1)^{2} - 8k(2s^{2} - s - 2)} \rceil}\) .  相似文献   

12.
We consider a quasi-variational inequality (q.v.i.) introduced by A. Friedman and D. Kinderlehrer. A q.v.i. of this form gives rise, at least formally, to a Stefan problem of melting of water, where the relation ?vx(x, t) = ?a(x, t)·(t) + b(x, t) holds on the free boundary x = s(t), and a > 0, b ? 0; the water temperature, v(x, t), is not necessarily nonnegative. In the standard Stefan problem a ≡ 1, b ≡ 0, and v ? 0. Friedman and Kinderlehrer proved the existence of a solution of the q.v.i. by a fixed point theorem for monotone mappings. Here we prove the existence of a solution by an entirely different method, based on finite difference approximations. The solution is shown to be smoother than that constructed by Friedman and Kinderlehrer.  相似文献   

13.
A t-(v, k, λ) covering design is a pair (X, B) where X is a v-set and B is a collection of k-sets in X, called blocks, such that every t element subset of X is contained in at least λ blocks of B. The covering number, Cλ(t, k, v), is the minimum number of blocks a t-(v, k, λ) covering design may have. The chromatic number of (X, B) is the smallest m for which there exists a map φ: XZm such that ∣φ((β)∣ ≥2 for all β ∈ B, where φ(β) = {φ(x): x ∈ β}. The system (X, B) is equitably m-chromatic if there is a proper coloring φ with minimal m for which the numbers ∣φ?1(c)∣ cZm differ from each other by at most 1. In this article we show that minimum, (i.e., ∣B∣ = C λ (t, k, v)) equitably 3-chromatic 3-(v, 4, 1) covering designs exist for v ≡ 0 (mod 6), v ≥ 18 for v ≥ 1, 13 (mod 36), v ≡ 13 and for all numbers v = n, n + 1, where n ≡ 4, 8, 10 (mod 12), n ≥ 16; and n = 6.5a 13b 17c ?4, a + b + c > 0, and n = 14, 62. We also show that minimum, equitably 2-chromatic 3-(v, 4, 1) covering designs exist for v ≡ 0, 5, 9 (mod 12), v ≥ 0, v = 2.5a 13b 17c + 1, a + b + c > 0, and v = 23. © 1993 John Wiley & Sons, Inc.  相似文献   

14.
Let SSR(v, 3) denote the set of all integer b* such that there exists a RTS(v, 3) with b* distinct triples. In this paper, we determine the set SSR(v, 3) for v ≡ 3 (mod 6) and v ≥ 3 with only five undecided cases. We establish that SSR(v, 3) = P(v, 3) for v ≡ 3 (mod 6), v ≥ 21 and v ≠ 33, 39 where P(v, 3) = {mv, mv + 4, mv + 6, mv + 7, …, 3mv} and mv, = v(v ? 1)/6. As a by‐product, we remove the last two undecided cases for the intersection numbers of Kirkman triple system of order 27, this improves the known result provided in [ 2 ]. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 275–289, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10037  相似文献   

15.
Let S(n, k, v) denote the number of vectors (a0,…, an?1) with nonnegative integer components that satisfy a0 + … + an ? 1 = k and Σi=0n?1iaiv (mod n). Two proofs are given for the relation S(n, k, v) = S(k, n, v). The first proof is by algebraic enumeration while the second is by combinatorial construction.  相似文献   

16.
Let p be a prime and let b be a positive integer. If a (v, k, λ, n) difference set D of order n = p b exists in an abelian group with cyclic Sylow p-subgroup S, then \({p\in\{2,3\}}\) and |S| = p. Furthermore, either p = 2 and vλ ≡ 2 (mod 4) or the parameters of D belong to one of four families explicitly determined in our main theorem.  相似文献   

17.
In this paper, we look at the existence of (v K) pairwise balanced designs (PBDs) for a few sets K of prime powers ≥ 8 and also for a number of subsets K of {5, 6, 7, 8, 9}, which contain {5}. For K = {5, 7}, {5, 8}, {5, 7, 9}, we reduce the largest v for which a (v, K)‐PBD is unknown to 639, 812, and 179, respectively. When K is Q≥8, the set of all prime powers ≥ 8, we find several new designs for 1,180 ≤ v ≤ 1,270, and reduce the largest unsolved case to 1,802. For K =Q0,1,5(8), the set of prime powers ≥ 8 and ≡ 0, 1, or 5 (mod 8) we reduce the largest unknown case from 8,108 to 2,612. We also obtain slight improvements when K is one of {8, 9} or Q0,1(8), the set of prime powers ≡ 0 or 1 (mod 8). © 2004 Wiley Periodicals, Inc.  相似文献   

18.
L. Ji 《组合设计杂志》2004,12(2):92-102
Let B3(K) = {v:? an S(3,K,v)}. For K = {4} or {4,6}, B3(K) has been determined by Hanani, and for K = {4, 5} by a previous paper of the author. In this paper, we investigate the case of K = {4,5,6}. It is easy to see that if vB3 ({4, 5, 6}), then v ≡ 0, 1, 2 (mod 4). It is known that B3{4, 6}) = {v > 0: v ≡ 0 (mod 2)} ? B3({4,5,6}) by Hanani and that B3({4, 5}) = {v > 0: v ≡ 1, 2, 4, 5, 8, 10 (mod 12) and v ≠ 13} ? B3({4, 5, 6}). We shall focus on the case of v ≡ 9 (mod 12). It is proved that B3({4,5,6}) = {v > 0: v ≡ 0, 1, 2 (mod 4) and v ≠ 9, 13}. © 2003 Wiley Periodicals, Inc.  相似文献   

19.
Let D(v) denote the maximum number of pairwise disjoint Steiner triple systems of order v. In this paper, it is proved that if D(2 + n) = n, p is a prime number, p ≡ 7 (mod 8) or p? {5, 17, 19, 2}, and (p, n) ≠ (5, 1), then D(2 + pn) = pn.  相似文献   

20.
On the interval [t 0, ∞), we consider the following group pursuit problem with one evader: 1 $$ z_i^{(l)} + a_1 (t)z_i^{(l - 1)} + a_2 (t)z_i^{(l - 2)} + \cdots + a_l (t)z_i = u_i - v, u_i ,v \in V, z_i^{(q)} (t_0 ) = z_i^q , $$ where z i , u i , vR v , (v ≥ 2), V is a strictly convex compact set in R v , the functions a 1(t), a 2(t), …, a l (t) are continuous, i = 1, 2, …, n and q = 0, 1, …, l ? 1. Let ? q (t, s) be the solution of the Cauchy problem $$ \begin{gathered} \omega ^{(l)} + a_1 (t)\omega ^{(l - 1)} + a_2 (t)\omega ^{(l - 2)} + \cdots + a_l (t)\omega = 0, \omega ^{(q)} (s) = 1, \hfill \\ \omega ^{(r)} (s) = 0, r = 0, \ldots q - 1,q + 1, \ldots ,l - 1, \hfill \\ \end{gathered} $$ and let $$ \xi _\iota (t) = \varphi _0 (t,t_0 )Z_i^0 + \varphi _1 (t,t_0 )Z_i^1 + \cdots + \varphi _{l - 1} (t,t_0 )Z_i^{l - 1} . $$ We prove that if there exist continuous functions α i (t) and ξ i 1 (t) such that the ξ i 1 (t) are Bohr almost periodic on [t 0, ∞), α i (t) > 0 for all tt 0, lim t→∞(ξ i 1 (t) ? α i (t)ξ i (t)) = 0, lim t→∞(min i α i (t) ∝ t0 t |? l?1(t, s)| ds) = ∞, and there exist points h i 0 H i 1 = {ξ i 1 (t), t ∈ [0, ∞)} such that 0 ∈ Int co{h i 0 }, then the pursuit problem with evader discrimination is solvable.  相似文献   

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