RHPMDs and FHPMDs with k = 4 and Type 4 u

A (v, k, 1)-HPMD is called a frame (briefly, k-FHPMD), if the blocks of the HPMD can be partitioned into v partial parallel classes such that the complement of each partial parallel class is a group of the HPMD. A (v, k, 1)-HPMD is called resolvable (briefly, k-RHPMD), if the blocks of the HPMD can be partitioned into parallel classes. In this article, (i) we shall construct 3-FHPMDs of type 3⁶ and 21⁶ to completely settle the existence of 3-FHPMD of type h^u; (ii) we shall show that the necessary conditions for the existence of 4-FHPMD of type h^u are sufficient for the case h = 4; (iii) we shall show that the necessary conditions for the existence of 4-RHPMD of type h^u are sufficient for the case h = 4.     

Self-converse Mendelsohn designs with block size 4t + 2

A Mendelsohn design MD(v, k, λ) is a pair (X, B) where X is a v-set together with a collection B of cyclic k-tuples from X such that each ordered pair from X, as adjacent entries, is contained in exactly λk-tuples of B. An MD(v, k, λ) is said to be self-converse, denoted by SCMD(v, k, λ) = (X, B, f), if there is an isomorphic mapping from (X, B) to (X, B⁻¹), where B⁻¹ = {B⁻¹ = 〈x_k, x_k−1, … x₂, x₁〉; B = 〈x₁, … ,x_k〉 ∈ B.}. The existence of SCMD(v, 3, λ) and SCMD(v, 4, 1) has been settled by us. In this article, we will investigate the existence of SCMD(v, 4t + 2, 1). In particular, when 2t + 1 is a prime power, the existence of SCMD(v, 4t + 2, 1) has been completely solved, which extends the existence results for MD(v, k, 1) as well. © 1999 John Wiley & Sons, Inc. J. Combin Designs 7: 283–310, 1999     

3-(v, 4, 1) covering designs with chromatic numbers 2 and 3

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, C_λ(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: X → Z_m 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)∣ c ∈ Z_m 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 = 6.5^a 13^b 17^c ?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 = 2.5^a 13^b 17^c + 1, a + b + c > 0, and v = 23. © 1993 John Wiley & Sons, Inc.     

Upper bounds on the general covering number Cλ(v,k, t,m)

A collection of k‐subsets (called blocks) of a v‐set X (v) = {1, 2,…, v} (with elements called points) is called a t‐(v, k, m, λ) covering if for every m‐subset M of X (v) there is a subcollection of with such that every block K ∈ has at least t points in common with M. It is required that v ≥ k ≥ t and v ≥ m ≥ t. The minimum number of blocks in a t‐(v, k, m, λ) covering is denoted by C_λ(v, k, t, m). We present some constructions producing the best known upper bounds on C_λ(v, k, t, m) for k = 6, a parameter of interest to lottery players. © 2004 Wiley Periodicals, Inc.     

Self‐converse Mendelsohn designs with block size 4 and 5

A (v, k, λ)‐Mendelsohn design(X, ℬ︁) is called self‐converse if there is an isomorphic mapping ƒ from (X, ℬ︁) to (X, ℬ︁⁻¹), where ℬ︁⁻¹ = {B⁻¹ = 〈x_k, x_k−1,…,x₂, x₁〉: B = 〈x₁, x₂,…,x_k−1, x_k〉 ϵ ℬ︁}. In this paper, we give the existence spectrum for self‐converse (v, 4, 1)– and (v, 5, 1)– Mendelsohn designs. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 411–418, 2000     

Existence of HPMDs with block size five

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     

Covering pairs by quintuples with index λ  0 (mod 4)

A (v, k. λ) covering design of order v, block size k, and index λ is a collection of k-element subsets, called blocks, of a set V such that every 2-subset of V occurs in at least λ blocks. The covering problem is to determine the minimum number of blocks, α(v, k, λ), in a covering design. It is well known that $ \alpha \left({\nu,\kappa,\lambda } \right) \ge \left\lceil {\frac{\nu}{\kappa}\left\lceil {\frac{{\nu - 1}}{{\kappa - 1}}\lambda} \right\rceil} \right\rceil = \phi \left({\nu,\kappa,\lambda} \right) $, where [χ] is the smallest integer satisfying χ ≤ χ. It is shown here that α (v, 5, λ) = ?(v, 5, λ) + ? where λ ≡ 0 (mod 4) and e= 1 if λ (v?1)≡ 0(mod 4) and λv (v?1)/4 ≡ ?1 (mod 5) and e= 0 otherwise With the possible exception of (v,λ) = (28, 4). © 1993 John Wiley & Sons, Inc.     

On the Travelling Salesperson Problem in Many Dimensions

Consider d ? 2, and m points X₁, …, X_m that are independent uniformly distributed in [0, 1]^d. It is well known that the length T_m of the shortest tour through X₁, …, X_m satisfies lim_m→∞ E(T_m)/m^1?1/d = β(d) for a certain number β(d). We show that for some numerical constant K, .     

Subsets of a Hilbert space,having finite hausdorff dimension

Let X₂, X₂ be Hilbert spaces, X₂ X₁, X₂ is dense in X₁, the imbedding is compact,m X₂, dim_H ⁱ m and h⁽ⁱ⁾(m) are the Hausdorff dimension and the limit capacity (information dimension) of the setm with respect to the metrics of the spaces X_i (i=1, 2). Two examples are constructed. 1) An example of a setm bounded in X₂, such that: a) h⁽¹⁾(m) < (and, consequently, dim_H ¹ m); b)m cannot be covered by a countable collection of sets, compact in X₂ (and, consequently, dim_H ² m=). 2) an Example of a setm, compact in X₂, such that h⁽¹⁾(m) < and h⁽²⁾(m)=.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 163, pp. 154–165, 1987.     

On large sets of almost Hamilton cycle decompositions

A large set of CS(v, k, λ), k‐cycle system of order v with index λ, is a partition of all k‐cycles of K_v into CS(v, k, λ)s, denoted by LCS(v, k, λ). A (v ? 1)‐cycle is called almost Hamilton. The completion of the existence spectrum for LCS(v, v ? 1, λ) only depends on one case: all v ≥ 4 for λ = 2. In this article, it is shown that there exists an LCS(v, v ? 1,2) for any v ≡ 0,1 (mod 4) except v = 5, and for v = 6,7,10,11. © 2006 Wiley Periodicals, Inc. J Combin Designs 16: 53–69, 2008     

The intersection problem for m-cycle systems

Let I_m(v) denote the set of integers k for which a pair of m-cycle systems of K_v, exist, on the same vertex set, having k common cycles. Let J_m(v) = {0, 1, 2,…, t_v ?2, t_v} where t_v = v(v ? 1)/2m. In this article, if 2mn + x is an admissible order of an m-cycle system, we investigate when I_m(2mn + x) = J_m(2mn + x), for both m even and m odd. Results include J^m(2mn + 1) = I^m(2mn + 1) for all n > 1 if m is even, and for all n > 2 if n is odd. Moreover, the intersection problem for even cycle systems is completely solved for an equivalence class x (mod 2m) once it is solved for the smallest in that equivalence class and for K_2m+1. For odd cycle systems, results are similar, although generally the two smallest values in each equivalence class need to be solved. We also completely solve the intersection problem for m = 4, 6, 7, 8, and 9. (The cased m = 5 was done by C-M. K. Fu in 1987.) © 1993 John Wiley & Sons, Inc.     

The spectrum of additive BIB designs

In this article, the existence of additive BIB designs is discussed with direct and recursive constructions, together with investigation of a property of resolvability. Such designs can be used to construct infinite families of BIB designs. In particular, we obtain a series of B(sⁿ, ts^m, λt (ts^m ? 1) (s^n‐m ? 1)/[2(s^m ? 1)]) for any positive integer λ, such that sⁿ (sⁿ ? 1) λ ≡ 0 (mod s^m (s^m ? 1) and for any positive integer t with 2 ≤ t ≤ s^n‐m, where s is an odd prime power. Connections between additive BIB designs and other combinatorial objects such as multiply nested designs and perpendicular arrays are discussed. A construction of resolvable BIB designs with v = 4k is also proposed. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 235–254, 2007     

New upper bounds on the minimum size of covering designs

Let D = {B₁, B₂,…, B_b} be a finite family of k-subsets (called blocks ) of a v-set X(v) = {1, 2,…, v} (with elements called points ). Then D is a (v, k, t) covering design or covering if every t-subset of X(v) is contained in at least one block of D. The number of blocks, b, is the size of the covering, and the minimum size of the covering is called the covering number , denoted C(v, k, t). This article is concerned with new constructions of coverings. The constructions improve many upper bounds on the covering number C(v, k, t) © 1998 John Wiley & Sons, Inc. J Combin Designs 6:21–41, 1998     

Minimum norm problems over transportation polytopes

We consider the problem of updating input-output matrices, i.e., for given (m,n) matrices A ? 0, W ? 0 and vectors u ? $\text{R}$^m, v?$\text{R}$ⁿ, find an (m,n) matrix X ? 0 with prescribed row sums Σⁿ_j=1X_ij = u_i (i = 1,…,m) and prescribed column sums Σ^m_i=1X_ij = v_j (j = 1,…,n) which fits the relations X_ij = A_ij + λ_iW_ij + W_ij + W_ijμ_j for all i,j and some λ?$\text{R}$^m, μ?$\text{R}$ⁿ. Here we consider the question of existence of a solution to this problem, i.e., we shall characterize those matrices A, W and vectors u,v which lead to a solvable problem. Furthermore we outline some computational results using an algorithm of [2].     

A new class for large sets of almost Hamilton cycle decompositions

A k-cycle system of order v with index λ, denoted by CS(v, k, λ), is a collection A of k-cycles (blocks) of K _v such that each edge in K _v appears in exactly λ blocks of A. A large set of CS(v, k, λ)s is a partition of the set of all k-cycles of K _v into CS(v, k, λ)_s, and is denoted by LCS(v, k, λ). A (v ?1)-cycle in K _v is called almost Hamilton. The completion of the existence problem for LCS(v, v?1, λ) depends only on one case: all v ≥ 4 for λ = 2. In this paper, it is shown that there exists an LCS(v, v ? 1, 2) for all v ≡ 2 (mod 4), v ≥ 6.     

Direct construction methods for incomplete perfect Mendelsohn designs with block size four

Let v, k, λ, and n be positive integers. An incomplete perfect Mendelsohn design, denoted by (v,n,k,λ)-IPMD, is a triple (X,Y,B) where X is a v-set (of points), Y is an n-subset of X, and B is a collection of cyclically ordered k-subsets of X (called blocks) such that every ordered pair (a,b) E (X × X)\(Y × Y) appears t-apart in exactly λ blocks of B and no ordered pair (a,b) E Y × Y appears in any block of B for any t, where 1 ≤ t ≤ k − 1. In this article, we introduce an effective and easy way to construct IPMDs for k = 4 and even v − n, and use it to construct some small examples for λ = 1 and 2. Obviously, these results will play an important role to completely solve the existence of (v,n,4,λ)-IPMDs. Furthermore, we also use this method to construct some small examples for HPMDs. © 1996 John Wiley & Sons, Inc.     

A bound for wilson's theorem (I)

In 1975, Richard M. Wilson proved: Given any positive integers k ? 3 and λ, there exists a constant v⁰ = v⁰(k, λ) such that v ? B(k,λ) for every integer v ? v⁰ that satisfies λ(v ? 1) ≡ 0(mod k ? 1) and λv(v ? 1) ≡ 0[mod k(k ? 1)]. The proof given by Wilson does not provide an explicit value of v⁰. We try to find such a value v⁰(k, λ). In this article we consider the case λ = 1 and v ≡ 1[mod k(k ? 1)]. We show that: if k ? 3 and v = 1[mod k(k ? 1)] where v > k^kk5, then a B(v,k, 1) exists. © 1995 John Wiley & Sons, Inc.     

Nonzero decomposable symmetrized tensors

Suppose k₁ ? ? ? k_t ? 1, m₁ ? ?? m_r ? 1, k₁+ ? +k_t = m₁+ ? +m_r = m. Let λ=(k₁,…,k_t) be a character of the symmetric group S_m. The restriction of λ to S_m1X…XS_mr contains the principal character as a component if and only if λ majorizes (m₁,…,m_r). This result is used to characterize the index set of the nonzero decomposable symmetrized tensors, corresponding to S_m and λ, which are induced from a basis of the underlying vector space.     

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.