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1.
A construction is given for a (p2a(p+1),p2,p2a+1(p+1),p2a+1,p2a(p+1)) (p a prime) divisible difference set in the group H×Z2pa+1 where H is any abelian group of order p+1. This can be used to generate a symmetric semi-regular divisible design; this is a new set of parameters for λ1≠0, and those are fairly rare. We also give a construction for a (pa−1+pa−2+…+p+2,pa+2, pa(pa+pa−1+…+p+1), pa(pa−1+…+p+1), pa−1(pa+…+p2+2)) divisible difference set in the group H×Zp2×Zap. This is another new set of parameters, and it corresponds to a symmetric regular divisible design. For p=2, these parameters have λ12, and this corresponds to the parameters for the ordinary Menon difference sets.  相似文献   

2.
Let p be an odd prime, and D2p =<a, b|ap = b2 = 1, bab = a-1 the dihedral group of order 2p. In this paper, we completely classify the cubic Cayley graphs on D2p up to isomorphism by means of spectral method. By the way, we show that two cubic Cayley graphs on D2p are isomorphic if and only if they are cospectral. Moreover, we obtain the number of isomorphic classes of cubic Cayley graphs on D2p by using Gauss' celebrated law of quadratic reciprocity.  相似文献   

3.
Let G be a solvable block transitive automorphism group of a 2−(v,5,1) design and suppose that G is not flag transitive. We will prove that
(1) if G is point imprimitive, then v=21, and GZ21:Z6;
(2) if G is point primitive, then GAΓL(1,v) and v=pa, where p is a prime number with p≡21 (mod 40), and a an odd integer.
  相似文献   

4.
Knödel graphs form a class of bipartite incident-graph of circulant digraphs. This class has been extensively studied for the purpose of fast communications in networks, and it has deserved a lot of attention in this context. In this paper, we show that there exists an O(n log5 n)-time algorithm to recognize Knödel graphs of order 2n. The algorithm is based on a characterization of the cycles of length six in these graphs (bipartite incident-graphs of circulant digraphs always have cycles of length six). A consequence of our result is that the circulant digraphs whose chords are the power of two minus one can be recognized in O(n log5 n) time.  相似文献   

5.
For a 1-dependent stationary sequence {Xn} we first show that if u satisfies p1=p1(u)=P(X1>u)0.025 and n>3 is such that 88np131, then
P{max(X1,…,Xn)u}=ν·μn+O{p13(88n(1+124np13)+561)}, n>3,
where
ν=1−p2+2p3−3p4+p12+6p22−6p1p2,μ=(1+p1p2+p3p4+2p12+3p22−5p1p2)−1
with
pk=pk(u)=P{min(X1,…,Xk)>u}, k1
and
|O(x)||x|.
From this result we deduce, for a stationary T-dependent process with a.s. continuous path {Ys}, a similar, in terms of P{max0skTYs<u}, k=1,2 formula for P{max0stYsu}, t>3T and apply this formula to the process Ys=W(s+1)−W(s), s0, where {W(s)} is the Wiener process. We then obtain numerical estimations of the above probabilities.  相似文献   

6.
It is known that for sufficiently large n and m and any r the binomial coefficient (nm) which is close to the middle coefficient is divisible by pr where p is a ‘large’ prime. We prove the exact divisibility of (nm) by pr for p> c(n). The lower bound is essentially the best possible. We also prove some other results on divisibility of binomial coefficients.  相似文献   

7.
For a double array {V_(m,n), m ≥ 1, n ≥ 1} of independent, mean 0 random elements in a real separable Rademacher type p(1 ≤ p ≤ 2) Banach space and an increasing double array {b_(m,n), m ≥1, n ≥ 1} of positive constants, the limit law ■ and in L_p as m∨n→∞ is shown to hold if ■ This strong law of large numbers provides a complete characterization of Rademacher type p Banach spaces. Results of this form are also established when 0 p ≤ 1 where no independence or mean 0 conditions are placed on the random elements and without any geometric conditions placed on the underlying Banach space.  相似文献   

8.
In this paper it is proved that the exponential generating function of the numbers, denoted by N(p, q), of irreducible coverings by edges of the vertices of complete bipartite graphs Kp,q equals exp(xey + yexxyxy) − 1.  相似文献   

9.
We consider a family of second-order elliptic operators {L_ε} in divergence form with rapidly oscillating and periodic coefficients in Lipschitz and convex domains in R~n. We are able to show that the uniform W~(1,p) estimate of second order elliptic systems holds for 2n/(n+1)-δ p 2n/(n-1)+ δ where δ 0 is independent of ε and the ranges are sharp for n = 2, 3. And for elliptic equations in Lipschitz domains, the W~(1,p) estimate is true for 3/2-δ p 3 + δ if n ≥ 4, similar estimate was extended to convex domains for 1 p ∞.  相似文献   

10.
For the pth-order linear ARCH model,
, where 0 > 0, i 0, I = 1, 2, …, p, {t} is an i.i.d. normal white noise with Et = 0, Et2 = 1, and t is independent of {Xs, s < t}, Engle (1982) obtained the necessary and sufficient condition for the second-order stationarity, that is, 1 + 2 + ··· + p < 1. In this note, we assume that t has the probability density function p(t) which is positive and lower-semicontinuous over the real line, but not necessarily Gaussian, then the geometric ergodicity of the ARCH(p) process is proved under Et2 = 1. When t has only the first-order absolute moment, a sufficient condition for the geometric ergodicity is also given.  相似文献   

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