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1.
Let P be the Petersen graph, and K
u(h) the complete multipartite graph with u parts of size h. A decomposition of K
u(h) into edge-disjoint copies of the Petersen graph P is called a P-decomposition of K
u(h) or a P-group divisible design of type h
u
. In this paper, we show that there exists a P-decomposition of K
u(h) if and only if h2u(u-1) o 0 mod 30{h^2u(u-1)\equiv 0 \pmod {30}} , h(u-1) o 0 mod 3{h(u-1)\equiv 0\pmod 3} , and u ≥ 3 with a definite exception (h, u) = (1, 10). 相似文献
2.
For every positive integer n, consider the linear operator U n on polynomials of degree at most d with integer coefficients defined as follows: if we write ${\frac{h(t)}{(1 - t)^{d + 1}}=\sum_{m \geq 0} g(m) \, t^{m}}For every positive integer n, consider the linear operator U
n
on polynomials of degree at most d with integer coefficients defined as follows: if we write
\frach(t)(1 - t)d + 1=?m 3 0 g(m) tm{\frac{h(t)}{(1 - t)^{d + 1}}=\sum_{m \geq 0} g(m) \, t^{m}} , for some polynomial g(m) with rational coefficients, then
\fracUnh(t)(1- t)d+1 = ?m 3 0g(nm) tm{\frac{{\rm{U}}_{n}h(t)}{(1- t)^{d+1}} = \sum_{m \geq 0}g(nm) \, t^{m}} . We show that there exists a positive integer n
d
, depending only on d, such that if h(t) is a polynomial of degree at most d with nonnegative integer coefficients and h(0) = 1, then for n ≥ n
d
, U
n
h(t) has simple, real, negative roots and positive, strictly log concave and strictly unimodal coefficients. Applications are
given to Ehrhart δ-polynomials and unimodular triangulations of dilations of lattice polytopes, as well as Hilbert series
of Veronese subrings of Cohen–Macauley graded rings. 相似文献
3.
In this article, it is shown that the necessary condition for the existence of a holey perfect Mendelsohn design (HPMD) with block size 5 and type hn, namely, n ≥ 5 and n(n - 1)hn ≡ 0 (mod 5), is also sufficient, except possibly for a few cases. The results of this article guarantee the analogous existence results for group divisible designs (GDDs) of group-type hn with block size k = 5 and having index λ = 4. Moreover, some more conclusive results for the existence of (v, 5, 1)-perfect Mendelsohn designs (PMDs) are also mentioned. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 257–273, 1997 相似文献
4.
Let M = {m1, m2, …, mh} and X be a v-set (of points). A holey perfect Mendelsohn designs (briefly (v, k, λ) - HPMD), is a triple (X, H, B), where H is a collection of subsets of X (called holes) with sizes M and which partition X, and B is a collection of cyclic k-tuples of X (called blocks) such that no block meets a hole in more than one point and every ordered pair of points not contained in a hole appears t-apart in exactly λ blocks, for 1 ≤ t ≤ k − 1. The vector (m1, m2, …, mh) is called the type of the HPMD. If m1 = m2 = … = mh = m, we write briefly mh for the type. In this article, it is shown that the necessary condition for the existence of a (v, 4, λ) - HPMD of type mh, namely, is also sufficient with the exception of types 24 and 18 with λ = 1, and type m4 for odd m with odd λ. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 203–213, 1997 相似文献
5.
Changchun Liu 《Monatshefte für Mathematik》2012,94(3):237-249
In this paper, we study the initial-boundary value problem of porous medium equation ρ(x)u
t
= Δu
m
+ V(x)h(t)u
p
in a cone D = (0, ∞) × Ω, where V(x) ~ |x|s, h(t) ~ ts{V(x)\,{\sim}\, |x|^\sigma, h(t)\,{\sim}\, t^s}. Let ω
1 denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on Ω and let l denote the positive root of l
2 + (n − 2)l = ω
1. We prove that if
m < p £ 1+(m-1)(1+s)+\frac2(s+1)+sn+l{m < p \leq 1+(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}, then the problem has no global nonnegative solutions for any nonnegative u
0 unless u
0 = 0; if ${p >1 +(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}${p >1 +(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}, then the problem has global solutions for some u
0 ≥ 0. 相似文献
6.
Yun Qing Xu 《数学学报(英文版)》2009,25(8):1325-1336
A Latin squares of order v with ni missing sub-Latin squares (holes) of order hi (1 〈= i 〈 k), which are disjoint and spanning (i.e. ∑k i=l1 nihi = v), is called a partitioned incomplete Latin squares and denoted by PILS. The type of PILS is defined by (h1n1 h2n2…hknk ). If any two PILS inaset of t PILS of type T are orthogonal, then we denote the set by t-HMOLS(T). It has been proved that 3-HMOLS(2n31) exist for n ≥6 with 11 possible exceptions. In this paper, we investigate the existence of 3-HMOLS(2nu1) with u ≥ 4, and prove that 3-HMOLS(2~u1) exist if n ≥ 54 and n ≥7/4u + 7. 相似文献
7.
The generalized Petersen graph GP (n, k), n ≤ 3, 1 ≥ k < n/2 is a cubic graph with vertex-set {uj; i ? Zn} ∪ {vj; i ? Zn}, and edge-set {uiui, uivi, vivi+k, i?Zn}. In the paper we prove that (i) GP(n, k) is a Cayley graph if and only if k2 ? 1 (mod n); and (ii) GP(n, k) is a vertex-transitive graph that is not a Cayley graph if and only if k2 ? -1 (mod n) or (n, k) = (10, 2), the exceptional graph being isomorphic to the 1-skeleton of the dodecahedon. The proof of (i) is based on the classification of orientable regular embeddings of the n-dipole, the graph consisting of two vertices and n parallel edges, while (ii) follows immediately from (i) and a result of R. Frucht, J.E. Graver, and M.E. Watkins [“The Groups of the Generalized Petersen Graphs,” Proceedings of the Cambridge Philosophical Society, Vol. 70 (1971), pp. 211-218]. © 1995 John Wiley & Sons, Inc. 相似文献
8.
The possible continuation of solutions of the nonlinear heat equation in RN × R+ ut = Δum + up with m > 0, p > 1, after the blowup time is studied and the different continuation modes are discussed in terms of the exponents m and p. Thus, for m + p ≤ 2 we find a phenomenon of nontrivial continuation where the region {x : u(x, t) = ∞} is bounded and propagates with finite speed. This we call incomplete blowup. For N ≥ 3 and p > m(N + 2)/(N − 2) we find solutions that blow up at finite t = T and then become bounded again for t > T. Otherwise, we find that blowup is complete for a wide class of initial data. In the analysis of the behavior for large p, a list of critical exponents appears whose role is described. We also discuss a number of related problems and equations. We apply the same technique of analysis to the problem of continuation after the onset of extinction, for example, for the equation ut = Δum − up, m > 0. We find that no continuation exists if p + m ≤ 0 (complete extinction), and there exists a nontrivial continuation if p + m > 0 (incomplete extinction). © 1997 John Wiley & Sons, Inc. 相似文献
9.
P. Erdös 《Israel Journal of Mathematics》1963,1(3):156-160
Denote byG(n; m) a graph ofn vertices andm edges. We prove that everyG(n; [n
2/4]+1) contains a circuit ofl edges for every 3 ≦l<c
2
n, also that everyG(n; [n
2/4]+1) contains ak
e(u
n, un) withu
n=[c
1 logn] (for the definition ofk
e(u
n, un) see the introduction). Finally fort>t
0 everyG(n; [tn
3/2]) contains a circuit of 2l edges for 2≦l<c
3
t
2.
This work was done while the author received support from the National Science Foundation, N.S.F. G.88. 相似文献
10.
Shu-Yu Hsu 《manuscripta mathematica》2013,140(3-4):441-460
Let n ≥ 3, 0 < m ≤ (n ? 2)/n, p > max(1, (1 ? m)n/2), and ${0 \le u_0 \in L_{loc}^p(\mathbb{R}^n)}$ satisfy ${{\rm lim \, inf}_{R\to\infty}R^{-n+\frac{2}{1-m}} \int_{|x|\le R}u_0\,dx = \infty}$ . We prove the existence of unique global classical solution of u t = Δu m , u > 0, in ${\mathbb{R}^n \times (0, \infty), u(x, 0) = u_0(x)}$ in ${\mathbb{R}^n}$ . If in addition 0 < m < (n ? 2)/n and u 0(x) ≈ A|x|?q as |x| → ∞ for some constants A > 0, q < n/p, we prove that there exist constants α, β, such that the function v(x, t) = t α u(t β x, t) converges uniformly on every compact subset of ${\mathbb{R}^n}$ to the self-similar solution ψ(x, 1) of the equation with ψ(x, 0) = A|x|?q as t → ∞. Note that when m = (n ? 2)/(n + 2), n ≥ 3, if ${g_{ij} = u^{\frac{4}{n+2}}\delta_{ij}}$ is a metric on ${\mathbb{R}^n}$ that evolves by the Yamabe flow ?g ij /?t = ?Rg ij with u(x, 0) = u 0(x) in ${\mathbb{R}^n}$ where R is the scalar curvature, then u(x, t) is a global solution of the above fast diffusion equation. 相似文献
11.
In this paper, two new direct construction methods are given for holey self‐orthogonal Latin squares with a symmetric orthogonal mate (HSOLSSOMs). Some new HSOLSSOMs using already known methods are also given. The known existence results for HSOLSSOMs of types 1m u1 and hn are improved; for type 1m u1 there remain just four possible exceptions with u odd and 3 ≤ u ≤ 15; for type hn, there are just two possible exceptions remaining, for (h, n) = (6, 12) and (6, 18). As a byproduct, the known existence results for three holey mutually orthogonal Latin squares (3 HMOLS) are also improved. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 435–444, 2001 相似文献
12.
The Blow-up Locus of Heat Flows for Harmonic Maps 总被引:5,自引:0,他引:5
Abstract
Let M and N be two compact Riemannian manifolds. Let u
k
(x, t) be a sequence of strong stationary weak heat flows from M×R
+ to N with bounded energies. Assume that u
k→u weakly in H
1, 2(M×R
+, N) and that Σt is the blow-up set for a fixed t > 0. In this paper we first prove Σt is an H
m−2-rectifiable set for almost all t∈R
+. And then we prove two blow-up formulas for the blow-up set and the limiting map. From the formulas, we can see that if the
limiting map u is also a strong stationary weak heat flow, Σt is a distance solution of the (m− 2)-dimensional mean curvature flow [1]. If a smooth heat flow blows-up at a finite time, we derive a tangent map or a weakly
quasi-harmonic sphere and a blow-up set ∪t<0Σt× {t}. We prove the blow-up map is stationary if and only if the blow-up locus is a Brakke motion.
This work is supported by NSF grant 相似文献
13.
We show that the supremum norm of solutions with small initial data of the generalized Benjamin-Bona-Mahony equation ut-△ut=(b,▽u)+up(a,▽u)in x?Rn,n≥2, with integer p≥3 , decays to zero like t-2/3 if n=2 and like t-1+6, for any δ0, if n≥3, when t tends to infinity. The proofs of these results are based on an analysis of the linear equation ut-△=(b,▽u)) and the associated oscillatory integral which may have nonisolated stationary points of the phase function. 相似文献
14.
In this paper, we consider the following nonlinear fractional three-point boundary-value problem:
*20c D0 + a u(t) + f( t,u(t) ) = 0, 0 < t < 1, u(0) = u¢(0) = 0, u¢(1) = ò0h u(s)\textds, \begin{array}{*{20}{c}} {D_{0 + }^\alpha u(t) + f\left( {t,u(t)} \right) = 0,\,\,\,\,0 < t < 1,} \\ {u(0) = u'(0) = 0,\,\,\,\,u'(1) = \int\limits_0^\eta {u(s){\text{d}}s,} } \\ \end{array} 相似文献
15.
We consider a class of kernel estimators [^(t)]n,h\hat{\tau}_{n,h} of the tail index of a Pareto-type distribution, which generalizes and includes the classical Hill estimator [^(a)]n,k\hat{a}_{n,k}. It is well-known that [^(a)]n,k\hat{a}_{n,k} is a consistent estimator of the tail index if and only if k→ ∞ and k/n→0. Under suitable assumptions on the kernel, [^(t)] n,h\hat{\tau} _{n,h} is consistent whenever the bandwidth is taken to be a sequence of non-random numbers satisfying h
n
→0 and nh
n
→ ∞. We extend this result and prove the consistency uniformly over a certain range of bandwidths. This permits the treatment
of estimators of the tail index based upon data-dependent bandwidths, which are often used in practice. In the process, we
establish a uniform in bandwidth result for kernel-type regression estimators with a fixed design, which will likely be of
separate interest. 相似文献
16.
G. Gripenberg S.-O. Londen J. Prüss 《Mathematical Methods in the Applied Sciences》1997,20(16):1427-1448
It is proved that there is a (weak) solution of the equation ut=a*uxx+b*g(ux)x+f, on ℝ+ (where * denotes convolution over (−∞, t)) such that ux is locally bounded. Emphasis is put on having the assumptions on the initial conditions as weak as possible. The kernels a and b are completely monotone and if a(t)=t−α, b(t)=t−β, and g(ξ)∼sign(ξ)∣ξ∣γ for large ξ, then the main assumption is that α>(2γ+2)/(3γ+1)β+(2γ−2)/(3γ+1). © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd. 相似文献
17.
Asaf Nachmias 《Geometric And Functional Analysis》2009,19(4):1171-1194
Let {G n } be a sequence of finite transitive graphs with vertex degree d = d(n) and |G n | = n. Denote by p t (v, v) the return probability after t steps of the non-backtracking random walk on G n . We show that if p t (v, v) has quasi-random properties, then critical bond-percolation on G n behaves as it would on a random graph. More precisely, if $\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}$ then the size of the largest component in p-bond-percolation with ${p =\frac{1+O(n^{-1/3})}{d-1}}
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