Extending symmetric designs

If a symmetric 2-design with parameters (v, k, λ) is extendable, then one of the following holds: v = 4λ + 3, k = 2λ + 1; or v = (λ + 2)(λ² + 4λ + 2), k = λ² + 3λ + 1; or v = 111, k = 11, λ = 1; or v = 495, k = 39, λ = 3. In particular, there are at most three sets of extendable symmetric design parameters with any given value of λ. As a consequence, the only twice-extendable symmetric design is the 21-point projective plane.     

Resolvable balanced bipartite designs

A block B denotes a set of k = k₁ + k₂ elements which are divided into two subsets, B¹ and B², where ∣Bⁱ∣ = k_i, i = 1 or 2. Two elements are said to be linked in B if and only if they belong to different subsets of B. A balanced bipartite design, BBD(v, k₁, k₂, λ), is an arrangement of v elements into b blocks, each containing k elements such that each element occurs in exactly r blocks and any two distinct elements are linked in exactly λ blocks. A resolvable balanced bipartite design, RBBD(v, k₁, k₂, λ), is a BBD(v, k₁, k₂, λ), the b blocks of which can be divided into r sets which are called complete replications, such that each complete replication contains all the v elements of the design.Necessary conditions for the existence of RBBD(v, 1, k₂, λ) and RBBD(v, n, n, λ) are obtained and it is shown that some of the conditions are also sufficient. In particular, necessary and sufficient conditions for the existence of RBBD(v, 1, k₂, λ), where k₂ is odd or equal to two, and of RBBD(v, n, n, λ), where n is even and 2n ? 1 is a prime power, are given.     

Further results on the existence of nested orthogonal arrays

Nested orthogonal arrays provide an option for designing an experimental setup consisting of two experiments, the expensive one of higher accuracy being nested in a larger and relatively less expensive one of lower accuracy. We denote by OA_(λ, μ)(t, k, (v, w)) (or OA(t, k, (v, w)) if λ = μ = 1) a (symmetric) orthogonal array OA _λ (t, k, v) with a nested OA _μ (t, k, w) (as a subarray). It is proved in this article that an OA(t, t + 1,(v, w)) exists if and only if v ≥ 2w for any positive integers v, w and any strength t ≥ 2. Some constructions of OA_(λ, μ)(t, k, (v, w))′s with λ ≠ μ and k ? t > 1 are also presented.     

On three-designs of small order

For positive integers t?k?v and λ we define a t-design, denoted B_i[k,λ;v], to be a pair (X,B) where X is a set of points and B is a family, (B_i:i?I), of subsets of X, called blocks, which satisfy the following conditions: (i) |X|=v, the order of the design, (ii) |B_i|=k for each i?I, and (iii) every t-subset of X is contained in precisely λ blocks. The purpose of this paper is to investigate the existence of 3-designs with 3?k?v?32 and λ>0.Wilson has shown that there exists a constant N(t, k, v) such that designs B_t[k,λ;v] exist provided λ>N(t,k,v) and λ satisfies the trivial necessary conditions. We show that N(3,k,v)=0 for most of the cases under consideration and we give a numerical upper bound on N(3, k, v) for all 3?k?v?32. We give explicit constructions for all the designs needed.     

A symmetric design with parameters 2-(49, 16, 5)

A symmetric 2-design with parameters (v, k, λ) = (49, 16, 5) is constructed. Both this design and its residual, a design with parameters (v, b, r, k, λ) = (33, 48, 16, 11, 5), seem to be new. The derived designs do not have repeated blocks. The group of the design is cyclic of order 15. There is no polarity.     

Handcuffed designs

Handcuffed designs are a particular case of block designs on graphs. A handcuffed design with parametersv, k, λ consists of a system of orderedk-subsets of av-set, called handcuffed blocks. In a block {A ₁,A ₂,?, A _k } each element is assumed to be handcuffed to its neighbours and the block containsk ? 1 handcuffed pairs (A ₁,A ₂), (A ₂,A ₃), ? (A _k?1,A _k ). These pairs are considered unordered. The collection of handcuffed blocks constitute a hundcuffed design if the following are satisfied: (1) each element of thev-set appears amongst the blocks the same number of times (and at most once in a block) and (2) each pair of distinct elements of thev-set are handcuffed in exactly λ of the blocks. If the total number of blocks isb and each element appears inr blocks the following conditions are necessary for the handcuffed design to exist:

1. λv(v?1) = (k?1) b,
2. rv = kb.

We denote byH(v, k, λ) the class of all handcuffed designs with parametersv, k, λ and sayH (v, k, λ) exists if there is a design with parametersv, k, λ. In this paper we prove that the necessary conditions forH (v, k, λ) exist are also sufficient in the following cases: (a)λ = 1 or 2; (b)k = 3; (c)k is evenk = 2h, and (λ, 2h ? 1) = 1; (d)k is odd,k = 2h + 1, and (λ, 4h)=2 or (λ, 4h)=1.     

A few more Kirkman squares and doubly near resolvable BIBDs with block size 3

A Kirkman square with index λ, latinicity μ, block size k, and v points, KS_k(v;μ,λ), is a t×t array (t=λ(v-1)/μ(k-1)) defined on a v-set V such that (1) every point of V is contained in precisely μ cells of each row and column, (2) each cell of the array is either empty or contains a k-subset of V, and (3) the collection of blocks obtained from the non-empty cells of the array is a (v,k,λ)-BIBD. In a series of papers, Lamken established the existence of the following designs: KS₃(v;1,2) with at most six possible exceptions [E.R. Lamken, The existence of doubly resolvable (v,3,2)-BIBDs, J. Combin. Theory Ser. A 72 (1995) 50-76], KS₃(v;2,4) with two possible exceptions [E.R. Lamken, The existence of KS₃(v;2,4)s, Discrete Math. 186 (1998) 195-216], and doubly near resolvable (v,3,2)-BIBDs with at most eight possible exceptions [E.R. Lamken, The existence of doubly near resolvable (v,3,2)-BIBDs, J. Combin. Designs 2 (1994) 427-440]. In this paper, we construct designs for all of the open cases and complete the spectrum for these three types of designs. In addition, Colbourn, Lamken, Ling, and Mills established the spectrum of KS₃(v;1,1) in 2002 with 23 possible exceptions. We construct designs for 11 of the 23 open cases.     

Existence of resolvable optimal strong partially balanced designs

We shall refer to a strong partially balanced design SPBD(v,b,k;λ,0) whose b is the maximum number of blocks in all SPBD(v,b,k;λ,0), as an optimal strong partially balanced design, briefly OSPBD(v,k,λ). Resolvable strong partially balanced design was first formulated by Wang, Safavi-Naini and Pei [Combinatorial characterization of l-optimal authentication codes with arbitration, J. Combin. Math. Combin. Comput. 37 (2001) 205-224] in investigation of l-optimal authentication codes. This article investigates the existence of resolvable optimal strong partially balanced design ROSPBD(v,3,1). We show that there exists an ROSPBD(v,3,1) for any v?3 except v=6,12.     

Super-simple, Pan-orientable, and Pan-decomposable BIBDs with Block Size 4 and Related Structures

Let v, k, and μ be positive integers. A tournament T of order k, briefly k-tournament, is a directed graph on k vertices in which there is exactly one directed edge between any two vertices. A (v, k, λ = 2μ)-BIBD is called T-orientable if for each of its blocks B, it is possible to replace B by a copy of T on the set B so that every ordered pair of distinct points appears in exactly μ k-tournaments. A (v, k, λ = 2μ)-BIBD is called pan-orientable if it is T-orientable for every possible k-tournament T. In this paper, we continue the earlier investigations and complete the spectrum for (v, 4, λ = 2μ)-BIBDs which possess both the pan-orientable property and the pan-decomposable property first introduced by Granville et al. (Graphs Comb 5:57–61, 1989). For all μ, we are able to show that the necessary existence conditions are sufficient. When λ = 2 and v > 4, our designs are super-simple, that is they have no two blocks with more than two common points. One new corollary to this result is that there exists a (v, 4, 2)-BIBD which is both super-simple and directable for all v ≡ 1, 4 (mod 6), v > 4. Finally, we investigate the existence of pan-orientable, pan-decomposable (v, 4, λ = 2μ)-BIBDs with a pan-orientable, pan-decomposable (w, 4, λ = 2μ)-BIBD as a subdesign; here we obtain complete results for λ = 2, 4, but there remain several open cases for λ = 6 (mostly for v < 4w), and the case λ = 12 still has to be investigated.     

Resolvable path designs

A resolvable (balanced) path design, RBPD(v, k, λ) is the decomposition of λ copies of the complete graph on v vertices into edge-disjoint subgraphs such that each subgraph consists of $\text{v}\text{k}$ vertex-disjoint paths of length k ? 1 (k vertices). It is shown that an RBPD(v, 3, λ) exists if and only if v ≡ 9 (modulo 12/gcd(4, λ)). Moreover, the RBPD(v, 3, λ) can have an automorphism of order $\text{v}\text{3}$. For k > 3, it is shown that if v is large enough, then an RBPD(v, k, 1) exists if and only if v ≡ k² (modulo lcm(2k ? 2, k)). Also, it is shown that the categorical product of a k-factorable graph and a regular graph is also k-factorable. These results are stronger than two conjectures of P. Hell and A. Rosa     

A construction for PBIBD(2)'s

This note gives a new construction for PBIBD(2)'s that generalizes a construction of Hall's for finite projective planes, and that leads to a new PBIBD(2) with parameters (v, b, k, r, λ₁, λ₂) = (36, 60, 10, 0, 2).     

Block-transitive designs in affine spaces

This paper deals with block-transitive t-(v, k, λ) designs in affine spaces for large t, with a focus on the important index λ = 1 case. We prove that there are no non-trivial 5-(v, k, 1) designs admitting a block-transitive group of automorphisms that is of affine type. Moreover, we show that the corresponding non-existence result holds for 4-(v, k, 1) designs, except possibly when the group is 1-D affine. Our approach involves a consideration of the finite 2-homogeneous affine permutation groups.     

Partitioning the planes of AG2m(2) into 2-designs

A t-design S_λ(t, k, v) is an arrangement of v elements in blocks of k elements each such that every t element subset is contained in exactly λ blocks. A t-design S_λ(t, k, v) is called t′-resolvable if the blocks can be partitioned into families such that each family is the block system of a S_λ(t′, k, v). It is shown that the S₁(3, 4, 2^2m) design of planes on an even dimensional affine space over the field of two elements is 2-resolvable. Each S₁(2, 4, 2^2m) given by the resolution is itself 1-resolvable. As a corollary it is shown that every odd dimensional projective space over the field of two elements admits a 1-packing of 1-spreads, i.e. a partition of its lines into families of mutually disjoint lines whose union covers the space. This 1-packing may be generated from any one of its spreads by repeated application of a fixed collineation.     

Neighbor designs

A neighbor design is an arrangement of r copies of each of v varieties into b circular blocks of size k > 1 such that neighboring objects in each block are distinct and every pair of distinct varieties appears as neighbors in the set of circular blocks exactly λ times. Necessary conditions for the existence of a neighbor design with these parameters v, k, λ, r, b are that $\text{r =}\text{λ(v ? 1)}\text{2}$, and $\text{b =}\text{λv(v ? 1)}\text{2k}$ be integers for k > 2 and v > 2; and for k = 2 or v = 2, it is also necessary that λ be even or k be even, respectively. In this paper we show that these necessary conditions are also sufficient by giving a method to construct a neighbor design for all values of the parameters satisfying the necessary conditions.     

Handcuffed designs

Handcuffed designs are a particular case of block designs on graphs. A handcuffed design with parameters v, k, λ consists of a system of ordered k-subsets of a v-set, called handcuffed blocks. In a block {A₁, A₂,…, A_k} each element is assumed to be handcuffed to its neighbors and the block contains k ? 1 handcuffed pairs (A₁, A₂), (A₂, A₁), …, (A_k?1, A_k). These pairs are considered unordered. The collection of handcuffed blocks constitutes a handcuffed design if the following are satisfied: (1) each element of the v-set appears amongst the blocks the same number of times (and at most once in a block) and (2) each pair of distinct elements of the v-set are handcuffed in exactly λ of the blocks. If the total number of blocks is b and each element appears in r blocks the following conditions are necessary for the handcuffed design to exist. (1) λv (v ? 1) = (k ? 1)b. (2) rv = kb. In this paper it is shown that the necessary conditions are also sufficient.     

On Defining Sets of Full Designs with Block Size Three

A defining set of a t-(v, k, λ) design is a subcollection of its blocks which is contained in no other t-design with the given parameters, on the same point set. A minimal defining set is a defining set, none of whose proper subcollections is a defining set. The spectrum of minimal defining sets of a design D is the set {|M| | M is a minimal defining set of D}. We show that if a t-(v, k, λ) design D is contained in a design F, then for every minimal defining set d _D of D there exists a minimal defining set d _F of F such that \({d_D = d_F\cap D}\). The unique simple design with parameters \({{\left(v,k, {v-2\choose k-2}\right)}}\) is said to be the full design on v elements; it comprises all possible k-tuples on a v set. Every simple t-(v, k, λ) design is contained in a full design, so studying minimal defining sets of full designs gives valuable information about the minimal defining sets of all t-(v, k, λ) designs. This paper studies the minimal defining sets of full designs when t = 2 and k = 3. Several families of non-isomorphic minimal defining sets of these designs are found. For given v, a lower bound on the size of the smallest and an upper bound on the size of the largest minimal defining set are given. The existence of a continuous section of the spectrum comprising approximately v values is shown, where just two values were known previously.     

Multiple solutions of coupled Sturm-Liouville systems

This paper deals with the coupled Sturm-Liouville system ? (pu′)′ + Pu + rv = λ₁u + λ₁N₁₁(u, v) + λ₂N₂₁(u, v), ? (qv′)′ + Qv + ru = λ₂v + λ₁N₁₂(u, v) + λ₂N₂₂(u, v), α₁₁u(0) + α₁₂u′(0) = 0 = α₂₁v(0) + α₂₂v′(0), β₁₁u(1) + β₁₂u′(1) = 0 = β₂₁v(1) + β₂₂v′(1). The functions p, P, q, Q, r are smooth; λ₁ and λ₂ are eigenparameters; N_ij(u, v) is analytic and of higher order. The linearized problem, all $\text{N}{}_{\mathrm{ij}}\text{\&z.tbnd; 0}$, is shown to have eigenvalues (λ₁, λ₂) which are continuously distributed along a sequence of monotonically decreasing curves in the λ₁λ₂-plane. A generalized Lyapunov-Schmidt method establishes that if (λ₁, λ₂) is near a simple eigenvalue of the linearized problem, then the number of small solutions of the nonlinear problem corresponds to the number of real roots of a certain polynomial.     

Threshold and generic type I behaviors for a supercritical nonlinear heat equation

We study blow-up of radially symmetric solutions of the nonlinear heat equation ut=Δu+|u|^p−1u either on RN or on a finite ball under the Dirichlet boundary conditions. We assume that N?3 and . Our first goal is to analyze a threshold behavior for solutions with initial data u₀=λv, where v∈C∩H¹ and v?0, v?0. It is known that there exists λ^?>0 such that the solution converges to 0 as t→∞ if 0<λ<λ^?, while it blows up in finite time if λ?λ^?. We show that there exist at most finitely many exceptional values λ₁=λ^?<λ₂λk such that, for all λ>λ^? with λ≠λj (j=1,2,…,k), the blow-up is complete and of type I with a flat local profile. Our method is based on a combination of the zero-number principle and energy estimates. In the second part of the paper, we employ the very same idea to show that the constant solution κ attains the smallest rescaled energy among all non-zero stationary solutions of the rescaled equation. Using this result, we derive a sharp criterion for no blow-up.     

Classification of amply regular graphs with <Emphasis Type="Italic">b</Emphasis><Subscript>1</Subscript> = 6

An undirected graph with v vertices in which the degrees of all vertices are equal to k, each edge is contained in exactly λ triangles, and the intersection of the neighborhoods of any two vertices at distance 2 contains exactly µ vertices is called amply regular with parameters (v, k, λ, µ). We complete the classification of amply regular graphs with b ₁ = 6, where b ₁ = k ? λ ? 1.     

A new strongly regular graph

A combinatorial construction is given of a strongly regular graph with parameters (v, k, λ, μ) = (280, 117, 44, 52) which was previously unknown.