Classification of some large sets and designs

In this paper, we briefly introduce an algorithm, based on the standard basis of trades, which has proven successful in the complete classification of certain combinatorial objects.     

New large sets of t‐designs

Abstact: We introduce generalizations of earlier direct methods for constructing large sets of t‐designs. These are based on assembling systematically orbits of t‐homogeneous permutation groups in their induced actions on k‐subsets. By means of these techniques and the known recursive methods we construct an extensive number of new large sets, including new infinite families. In particular, a new series of LS[3](2(2 + m), 8·3^m ? 2, 16·3^m ? 3) is obtained. This also provides the smallest known ν for a t‐(ν, k, λ) design when t ≥ 16. We present our results compactly for ν ≤ 61, in tables derived from Pascal's triangle modulo appropriate primes. © 2000 John Wiley & Sons, Inc. J Combin Designs 9: 40–59, 2001     

Further results on large sets of disjoint group-divisible designs

This paper is a continuation of a recent paper by Chen and Stinson, where some recursive constructions for large sets of group-divisible design with block size 3 are presented. In this paper, we give two new recursive constructions. In particular, we apply these constructions in the case of designs where every group has size 2.     

Representation sets for integral binary quadratic forms over the rationals

Let f be an integral binary form of discriminant d which represents n integrally. Two rational representations (r, s) and (r′, s′), with denominators prime to n, of n by f are called semiequivalent with respect to f if there is a rational automorph of f with determinant 1 and denominator m which takes (r, s) into (r′, s′) where (m, n) = 1 and m contains no factors p of d such that $\text{d}\text{p}{}^2$ is a discriminant. The number of such equivalence classes for a given f and n is sometimes finite. This number is obtained for forms with negative discriminants which have one class in each primitive genus.     

Some new large sets of t-designs

We investigate the existence of some large sets of size nine. We take advantage of the recursive and direct constructing method to show that in cases $LS\left[9\right](2,4,v)$ the trivial necessary condition is also sufficient. In particular $LS\left[9\right](2,4,29)$ is constructed. This fills a gap.     

The concept of a binary relation over partial enumerated sets

Translated fromAlgebra i Logika, Vol. 31, No. 3, pp. 304–316, May–June, 1992.     

On value sets of polynomials over a field

Let F be any field. Let p(F) be the characteristic of F if F is not of characteristic zero, and let p(F)=+∞ otherwise. Let A₁,…,A_n be finite nonempty subsets of F, and let

with k{1,2,3,…}, a₁,…,a_nF{0} and degg<k. We show that

When kn and |A_i|i for i=1,…,n, we also have

consequently, if nk then for any finite subset A of F we have

In the case n>k, we propose a further conjecture which extends the Erdős–Heilbronn conjecture in a new direction.     

A series of Siamese twin designs

A {0,±1}-matrix S is called a Siamese twin design sharing the entries of I if S=I+K−L, where I,K,L are nonzero {0,1}-matrices and both I+K and I+L are incidence matrices of symmetric designs with the same parameters. Let p and 2p+3 be prime powers and . We construct a Siamese twin design with parameters (4(p+1)²,2p²+3p+1,p²+p).     

Pure quantitative characterization of linear groups over the binary field

The purpose of this paper is to discuss the pure scalar characterization of the automorphism group Aut (L ₅(2)) and the linear group L ₆(2). It is proved that Aut(L ₅(2)) and L ₆(2) can be characterized quantitatively by the set of element orders. The main results are obtained by using William’s work on prime graph components of finite groups and Brauer characters in trivializing the possible 2-subgroups. __________ Translated from Chinese Annals of Mathematics, 2003, 24A(6): 675–682. This work was supported by the National Natural Science Foundation of China under grant number 10171074     

Hilbert series of subspace arrangements

The vanishing ideal I of a subspace arrangement V₁∪V₂∪?∪V_m⊆V is an intersection I₁∩I₂∩?∩I_m of linear ideals. We give a formula for the Hilbert polynomial of I if the subspaces meet transversally. We also give a formula for the Hilbert series of the product ideal J=I₁I₂?I_m without any assumptions about the subspace arrangement. It turns out that the Hilbert series of J is a combinatorial invariant of the subspace arrangement: it only depends on the intersection lattice and the dimension function. The graded Betti numbers of J are determined by the Hilbert series, so they are combinatorial invariants as well. We will also apply our results to generalized principal component analysis (GPCA), a tool that is useful for computer vision and image processing.     

A new algorithm for minimizing convex functions over convex sets

Let be a convex set for which there is an oracle with the following property. Given any pointz∈ℝ ⁿ the oracle returns a “Yes” ifz∈S; whereas ifz∉S then the oracle returns a “No” together with a hyperplane that separatesz fromS. The feasibility problem is the problem of finding a point inS; the convex optimization problem is the problem of minimizing a convex function overS. We present a new algorithm for the feasibility problem. The notion of a volumetric center of a polytope and a related ellipsoid of maximum volume inscribable in the polytope are central to the algorithm. Our algorithm has a significantly better global convergence rate and time complexity than the ellipsoid algorithm. The algorithm for the feasibility problem easily adapts to the convex optimization problem.     

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.     

A quantitative Khintchine-Groshev type theorem over a field of formal series

An asymptotic formula which holds almost everywhere is obtained for the number of solutions to theDiophantine inequalities ‖qA − p‖ < Ψ(‖g‖), where A is an n x m matrix (m > 1) over the field of formal Laurent series with coefficients from a finite field, and p and q are vectors of polynomials over the same finite field.     

New large sets of t‐designs with prescribed groups of automorphisms

In this article, we investigate the existence of large sets of 3‐designs of prime sizes with prescribed groups of automorphisms PSL(2,q) and PGL(2,q) for q < 60. We also construct some new interesting large sets by the use of the computer program DISCRETA. The results obtained through these direct methods along with known recursive constructions are combined to prove more extensive theorems on the existence of large sets. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 210–220, 2007     

Recursively enumerable sets of polynomials over a finite field

We prove that a relation over is recursively enumerable if and only if it is Diophantine over . We do this by first constructing a model of in , where n is represented by Zⁿ. In a second step, we show that it suffices to eliminate a bounded universal quantifier. Then finally, the hardest part of the proof is to show that we can eliminate this quantifier.