Kirkman covering designs with even-sized holes

A Kirkman holey covering design, denoted by KHCD(g^u), is a resolvable group-divisible covering design of type g^u. Each of its parallel class contains one block of size δ, while other blocks have size 3. Here δ is equal to 2, 3 and 4 when gu≡2, 3 and 4 (mod 3) in turn. In this paper, we study the existence problem of a KHCD(g^u) which has minimum possible number of parallel classes, and give a solution for most values of even g and u.     

The Existence of FrGBT D(4, g u )′ s

If the blocks of a GDD ${(X, \mathcal{G}, \mathcal{A})}$ with block size 4, index 3 and type g ^u can be arranged into a (gu)/4 × (gu) array, such that: (1) the main diagonal consists of u empty subarrays of size g/4 × g; (2) the blocks in each column form a partition of X\G for some ${G \in \mathcal{G}}$ , while the blocks in each row contains every element of X\G 3 times and no element of G for some ${G \in \mathcal {G}}$ , then the design is called a frame generalized balanced tournament design and denoted by FrGBT D(4, g ^u ). The necessary conditions for the existence of such a design are ${u \geq 6}$ and ${g \equiv 0}$ (mod 4). In this paper, the sufficiency of these conditions is proved with some possible exceptions.     

On the existence of (3,λ)‐semiframes of type 3u

A (k,λ)‐semiframe of type g^u is a (k,λ)‐group‐divisible design of type g^u (??, ??, ??), in which the collection of blocks ?? can be written as a disjoint union ??=??∪?? where ?? is partitioned into parallel classes of ?? and ?? is partitioned into holey parallel classes, each holey parallel class being a partition of ??\G_j for some G_j∈??. In this paper, we shall prove that the necessary conditions for (3,λ)‐semiframes of type 3^u are also sufficient with one exception. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 253–265, 2009     

Incomplete group divisible designs with block size four and general index

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.     

Kirkman Packing Designs KPD ({3, 5*},v)

A Kirkman packing design KPD ({3, 5^*},v) is a resolvable packing with maximum possible number of parallel classes, each class containing one block of size 5 and all other blocks of size three. Such designs can be used to construct certain threshold schemes in cryptography. In this paper, direct and recursive constructions are discussed for such designs. The existence of a KPD ({3, 5^*},v) for is established with a few possible exceptions.     

Up-embeddability via girth and the degree-sum of adjacent vertices

Let G be a simple graph of order n and girth g. For any two adjacent vertices u and v of G, if d _G (u) + d _G (v) ⩾ n − 2g + 5 then G is up-embeddable. In the case of 2-edge-connected (resp. 3-edge-connected) graph, G is up-embeddable if d _G (u) + d _G (v) ⩾ n − 2g + 3 (resp. d _G (u) + d _G (v) ⩾ n − 2g −5) for any two adjacent vertices u and v of G. Furthermore, the above three lower bounds are all shown to be tight. This work was supported by National Natural Science Foundation of China (Grant No. 10571013)     

The existence of large set of symmetric partitioned incomplete latin squares

In this paper, we investigate the existence of large sets of symmetric partitioned incomplete latin squares of type g^u (LSSPILSs) which can be viewed as a generalization of the well‐known golf designs. Constructions for LSSPILSs are presented from some other large sets, such as golf designs, large sets of group divisible designs, and large sets of Room frames. We prove that there exists an LSSPILS(g^u) if and only if u ≥ 3, g(u ? 1) ≡ 0 (mod 2), and (g, u) ≠ (1, 5).     

Convergence of an explicit finite volume scheme for first order symmetric systems

Summary.  This paper is devoted to the derivation of a O(h ^1/2) error estimate for the classical upwind, explicit in time, finite volume scheme for linear first order symmetric systems. Such a result already existed for the corresponding implicit in time finite volume scheme, since it can be interpreted as a particular case of the space-time discontinuous Galerkin method but the technique of proof, used in that case, does not extend to explicit schemes. The general framework, recently developed to analyse the convergence rate of finite volume schemes for non linear scalar conservation laws, can not be used either, because it is not adapted for systems, even linear. In this article, we propose a new technique, which takes advantage of the linearity of the problem. The first step consists in controlling the approximation error ∥u−u _h ∥ _L2 by an expression of the form <ν _h , g>−2<μ _h , gu>, where u is the exact solution, g is a particular smooth function, and μ _h , ν _h are some linear forms depending on the approximate solution u _h . The second step consists in carefully estimating the error terms <μ _h , gu> and <ν _h , g>, by using uniform stability results for the discrete problem and regularity properties of the continuous solution. Received December 20, 2001 / Revised version received January 2, 2001 / Published online November 27, 2002 Mathematics Subject Classification (1991): 65N30     

On Group-Divisible Designs with Block Size Four and Group-Type g u m 1

We investigate the spectrum for {4}-GDDs of type g ^u m ¹. Wedetermine, for each even g, all values of m for which a {4}-GDD of typeg ^u m ¹ exists, for every fourth value of u. We similarlydetermine, for each odd g 11 or 17, all values of m for which a {4}-GDD of typeg ^u m ¹ exists, for every third value of u. Finally, weestablish, up to a finite number of values of u, the spectrum for {4}-GDDs of typeg ^u m ¹ where gu is even, g {11, 17}.     

Modified frames with block size 3

A modified (k, λ)‐frame of type g^u is a modified (k, λ)‐GDD whose blocks can be partitioned into holey parallel classes, each of which is with respect to some group. Modified frames can be used to construct some other resolvable designs such as resolvable group divisible designs and semiframes. In this article, we shall investigate the existence of modified frames with block size 3. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 351–363, 2008     

Parabolic Schemes for Quasi-Linear Parabolic and Hyperbolic PDEs via Stochastic Calculus

We consider two quasi-linear initial-value Cauchy problems on ? ^d : a parabolic system and an hyperbolic one. They both have a first order non-linearity of the form φ(t, x, u)·?u, a forcing term h(t, x, u) and an initial condition u ₀ ∈ L ^∞(? ^d ) ∩ C ^∞(? ^d ), where φ (resp. h) is smooth and locally (resp. globally) Lipschitz in u uniformly in (t, x). We prove the existence of a unique global strong solution for the parabolic system. We show the existence of a unique local strong solution for the hyperbolic one and we give a lower bound regarding its blow up time. In both cases, we do not use weak solution theory but a direct construction based on parabolic schemes studied via a stochastic approach and a regularity result for sequences of parabolic operators. The result on the hyperbolic problem is performed by means of a non-classical vanishing viscosity method.     

The conformal plate buckling equation

The linear equation Δ²u = 1 for the infinitesimal buckling under uniform unit load of a thin elastic plate over ?² has the particularly interesting nonlinear generalization Δ_g²u = 1, where Δ_g = e^?2u Δ is the Laplace‐Beltrami operator for the metric g = e^2ug₀, with g₀ the standard Euclidean metric on ?². This conformal elliptic PDE of fourth order is equivalent to the nonlinear system of elliptic PDEs of second order Δu(x)+K_g(x) exp(2u(x)) = 0 and Δ K_g(x) + exp(2u(x)) = 0, with x ∈ ?², describing a conformally flat surface with a Gauss curvature function K_g that is generated self‐consistently through the metric's conformal factor. We study this conformal plate buckling equation under the hypotheses of finite integral curvature ∫ K_g exp(2u)dx = κ, finite area ∫ exp(2u)dx = α, and the mild compactness condition K₊ ∈ L¹(B₁(y)), uniformly w.r.t. y ∈ ?². We show that asymptotically for |x|→∞ all solutions behave like u(x) = ?(κ/2π)ln |x| + C + o(1) and K(x) = ?(α/2π) ln|x| + C + o(1), with κ ∈ (2π, 4π) and . We also show that for each κ ∈ (2π, 4π) there exists a K^* and a radially symmetric solution pair u, K, satisfying K(u) = κ and maxK = K^*, which is unique modulo translation of the origin, and scaling of x coupled with a translation of u. © 2001 John Wiley & Sons, Inc.     

The existence of uniform 5-GDDs

In this article, we construct group divisible designs (GDDs) with block size five, group-type g^u and index unity. The necessary condition for the existence of such a GDD is u ≷ 5, (u - 1)g ≡ 0 (mod 4) and u(u - 1)g² ≡ 0 (mod 20). It is shown that these necessary conditions are also sufficient, except possibly in a few cases. Additionally, a new construction to obtain GDDs using holey TDs is presented. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 275–299, 1997     

Generalized steiner triple systems with group size five

Generalized Steiner triple systems, GS(2, 3, n, g) are used to construct maximum constant weight codes over an alphabet of size g+1 with distance 3 and weight 3 in which each codeword has length n. The existence of GS(2, 3, n, g) has been solved for g=2, 3, 4, 9. In this paper, by introducing a special kind of holey generalized Steiner triple systems (denoted by HGS(2, 3, (n, u), g)), singular indirect product (SIP) construction for GDDs is used to construct generalized Steiner systems. The numerical necessary conditions for the existence of a GS(2, 3, n, g) are shown to be sufficient for g=5.     

The boundary trace and generalized boundary value problem for semilinear elliptic equations with coercive absorption

In the first part of the paper we establish the existence of a boundary trace for positive solutions of the equation ?Δu + g(x, u) = 0 in a smooth domain Ω ? ?^N, for a general class of positive nonlinearities. This class includes every space independent, monotone increasing g which satisfies the Keller‐Osserman condition as well as degenerate nonlinearities g_α,q of the form g_α,q (x, u) = d(x, ?Ω)^α |u|^q?1 u, with α > ?2 and q > 1. The boundary trace is given by a positive regular Borel measure which may blow up on compact sets. In the second part we concentrate on the family of nonlinearities {g_α,q}, determine the critical value of the exponent q (for fixed α > ?2) and discuss (a) positive solutions with an isolated singularity, for subcritical nonlinearities and (b) the boundary value problem for ?Δu + g_α,q (x, u) = 0 with boundary data given by a positive regular Borel measure (possibly unbounded). We show that, in the subcritical case, the problem possesses a unique solution for every such measure. © 2003 Wiley Periodicals, Inc.     

On a fractional partial differential equation with dominating linear part

It is proved that there is a (weak) solution of the equation u_t=a*u_xx+b*g(u_x)_x+f, on ℝ⁺ (where * denotes convolution over (−∞, t)) such that u_x 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.     

Petersen Graph Decompositions of Complete Multipartite Graphs

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).     

Truncated Transversal Designs: A New Lower Bound on the Number of Idempotent MOLS of Side n

A truncated transversal design TTD of type g^km¹ is a {k, k+1}-GDD of type g^km¹ in which each point on the group of size m lies only in blocks of size k+1. Thus a TTD of type g^km¹ is equivalent to a transversal design TD (k, g) having m disjoint parallel classes of blocks. We employ a new construction developed by the author (1993, J. Combin. Des.1, 15–26) to show that if g₁<g₂ and if there exists a TD (k, g₁) and a TD (k+1, g₂), then there exists a TTD of type (g₁g₂)^km¹ for any 0m(g₂ div g₁) g²₁. As a corollary, we obtain a new lower bound on the number of mutually orthogonal idempotent latin squares of side g: if g₁<g₂ and there exist r MOLS of side g₁ and r+1 MOLS of side g₂ , then N(1 ^g₁g₂)r.     

有向Kirkman填充设计的谱

§ 1　IntroductionLet X be a set of v points.A packing(directed packing) of X is a collection of subsets(ordered subsets) of X(called blocks) such that any pair(ordered pair) of distinct pointsfrom X occur together in atmostone block in the collection.A packing(directed packing)is called resolvable ifitsblock setadmitsa partition into parallel classes,each parallel classbeing a partition of the pointset X.A Kirkman triple system KTS(v) is a collection Tof3 -subsets of X(triples) suchthat …     

Gradient estimates for the elliptic and parabolic Lichnerowicz equations on compact manifolds

Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3. Denote D_g=-div_g?{\Delta_g=-{\rm div}_g\nabla} the Laplace–Beltrami operator. We establish some local gradient estimates for the positive solutions of the Lichnerowicz equation