The multiplier conjecture for elementary abelian groups

Applying the method that we presented in [19], in this article we prove: “Let G be an elementary abelian p-group. Let n = dn₁. If d(≠ p) is a prime not dividing n₁, and the order w of d mod p satisfies $ w > \frac{{d^2}}{3} $, then the Second Multiplier Theorem holds without the assumption n₁ > λ, except that only one case is yet undecided: w ≤ d², and $ \frac{{p - 1}}{{2w}} \ge 3 $, and t is a quadratic residue mod p, and t is not congruent to $ x^{\frac{{p - 1}}{{2w}}j} $ (mod p) (1 ≤ j < 2w), where t is an integer meeting the conditions of Second Multiplier Theorem, and x is a primitive root of p.”. © 1994 John Wiley & Sons, Inc.     

The multiplier conjecture for the case n = 4n1

In this article we prove that in the case n = 4n₁ if (v,2 · 7 · 31) = 1 and v is not divisible by 15, then the Second Multiplier Theorem holds without the assumption n₁ > λ. This improves a result due to McFarland. © 1995 John Wiley & Sons, Inc.     

McFarland's conjecture on abelian difference sets with multiplier −1

This paper is a continuation of the work by R.L. McFarland and S.L. Ma on abelian difference sets with –1 as a multiplier. More nonexistence results are obtained as a consequence of a theorem on the existence of sub-difference sets. In particular, nonexistence is shown for the two cases left undecided by McFarland and Ma.     

On the multiplier rules

We establish new results of first-order necessary conditions of optimality for finite-dimensional problems with inequality constraints and for problems with equality and inequality constraints, in the form of John’s theorem and in the form of Karush–Kuhn–Tucker’s theorem. In comparison with existing results, we weaken assumptions of continuity and of differentiability.     

Some results on the multiplier conjecture for the casen=5n 1

Using the method presented in [1], we obtain some new results which improve on the result of MacFarland's theorem (see [2]) in this case.     

On the Cameron-Praeger conjecture

This paper takes a significant step towards confirming a long-standing and far-reaching conjecture of Peter J. Cameron and Cheryl E. Praeger. They conjectured in 1993 that there are no non-trivial block-transitive 6-designs. We prove that the Cameron-Praeger conjecture is true for the important case of non-trivial Steiner 6-designs, i.e. for 6-(v,k,λ) designs with λ=1, except possibly when the group is PΓL(2,pe) with p=2 or 3, and e is an odd prime power.     

On the points-lines-planes conjecture

It has been conjectured that in any matroid, if W₁, W₂, W₃ denote the number of points, lines, and planes respectively, then W₂² ≥ W₁W₃. We prove this conjecture (and some strengthenings) for matroids in which no line has five or more points, thus generalizing a result of Stonesifer, who proved it for graphic matroids.     

Complete settling of the multiplier conjecture for the case of n=3pr

In this paper we improve the character approach to the multiplier conjecture that we presented after 1992, and thus we have made considerable progress in the case of n = 3n₁. We prove that in the case of n = 3n₁ Second multiplier theorem remains true if the assumption “n₁ > λ” is replaced by “(n₁, λ) = 1”. Consequentially we prove that if we let D be a (v, k, λ)-difference set in an abelian group G, and n = 3p^r for some prime p, (p,v) = 1, then p is a numerical multiplier of D.     

Complete settling of the multiplier conjecture for the case of n =3pr

In this paper we improve the character approach to the multiplier conjecture that we presented after 1992, and thus we have made considerable progress in the case of n=3n1. We prove that in the case of n = 3n1 Second multiplier theorem remains true if the assumption "n1 ＞λ" is replaced by "(n1, λ)=1".Consequentially we prove that if we let D be a (v, κ,λ)-difference set in an abelian group G, and n=3pr for some prime p, (p,v)=1, then p is a numerical multiplier of D.     

On the Vergne conjecture

Consider a Hamiltonian action by a compact Lie group on a possibly non-compact symplectic manifold. We give a short proof of a geometric formula for the decomposition into irreducible representations of the equivariant index of a \({{\mathrm{{{\mathrm{Spin}}}^c}}}\)-Dirac operator in this context. This formula was conjectured by Vergne in (Eur Math Soc Zürich I:635–664, 2007) and proved by Ma and Zhang in (Acta Math 212:11–57, 2014).     

On the direct sum conjecture

We prove the direct sum conjecture for various sets of systems of bilinear forms. Our results depend on a priori knowledge of the complexity of at least one of the direct summands and its underlying algebraic structure. We also briefly survey some previous results concerning the complexity and structure of minimal algorithms for various direct sum systems.     

On checking the Goldbach conjecture

This paper describes an attempt to check Goldbach's conjecture on a digital computer. The validity of the conjecture has been numerically verified up toN=33,000,000.     

On the second Goldbach conjecture

The results in this paper suggest that Goldbach's conjecture that every odd number is the sum of three primes is true even under the requirement that two of the primes be the same and the third be arbitrarily small.     

On the critical graph conjecture

Gol'dberg has recently constructed an infinite family of 3-critical graphs of even order. We now prove that if there exists a p(≥4)-critical graph K of odd order such that K has a vertex u of valency 2 and another vertex v ≠ u of valency ≤(p + 2)/2, then there exists a p-critical graph of even order.     

On Marcus-Wang conjecture

LetA _m ,B _m ,m=1, ...,p, be linear operators on ann-dimensional unitary space \(V.L = \sum\limits_{m = 1}^p {A_m \otimes B_m } \) is a linear operator on ?² V, the tensor product space with the customarily induced inner product. The numerical range ofL is defined as $$W\tfrac{1}{2}(L) = \left\{ {(L)x \otimes y,x \otimes y):x,y o.n.} \right\}$$ where “o.n.” means “orthonormal”. In [1], M.Marcus and B.Y. Wang conjecture: There exists no non-zero operatorL of minimum length less thann for whichW ₂ ¹ (L)=0. In this paper, we prove that this conjecture is true.