On collinear Griffiths points

J. Tabov has proved [1] that four Griffiths points are collinear if the vertices of a given quadrangle are on a circle. In this article we prove some generalization of this result in a very simple geometrical way (based on Desargues theorem). Received 9 July 1999; revised 13 December 1999.     

On the moduli space of quadruples of points in the boundary of complex hyperbolic space

We consider the space $ \mathcal{M} $ of ordered quadruples of distinct points in the boundary of complex hyperbolic n-space, $ H_\mathbb{C}^n $ , up to its holomorphic isometry group PU(n, 1): One of the important problems in complex hyperbolic geometry is to construct and describe a moduli space for $ \mathcal{M} $ . For n = 2, this problem was considered by Falbel, Parker, and Platis. The main purpose of this paper is to construct a moduli space for $ \mathcal{M} $ for any dimension n ≥ 1.     

On the space of quadruples of quinary alternating forms

We discuss some aspects of the invariant theory and arithmetic of the prehomogeneous vector space of quadruples of quinary alternating forms. In particular, we complete the explicit construction of all prehomogeneous covariants of this space and give the rational classification of maximal singular orbits.     

On quadruples of projectors connected by a linear relation

We describe the set of γ ∈ ℝ for which there exist quadruples of projectors P _i for a fixed collection of numbers α_i   ℝ₊, , such that α₁ P ₁ + α₂ P ₂ + α₃ P ₃ + α₃ P ₄ = γI. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 9, pp. 1289–1295, September, 2006.     

Characterization of regular Diophantine quadruples

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Characterization of regular Diophantine quadruples

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On the Griffiths group of the cubic sevenfold

Partially supported by Science Project Geometry of Algebraic Varieties, n. 0-198-SC1, and by fundings from M.U.R.S.T. and G.N.S.A.G.A. (C.N.R.), Italy     

Resolvable minimum coverings with quadruples

Let V be a finite set of v elements. A covering of the pairs of V by k-subsets is a family F of k-subsets of V, called blocks, such that each pair in V occurs in at least one member of F. For fixed v and k, the covering problem is to determine the number of blocks in any minimum covering. A minimum covering is resolvable if we can partition the blocks into classes (called resolution classes) such that every element is contained in precisely one block of each class. A resolvable minimum covering of the pairs of V by k-subsets is denoted by RC(v, k). In this article, we show that there exist RC(v, 4) for v ≡ 0 (mod 4) except for v = 12 and possibly for v ∈ {104, 108, 116, 132, 156, 164, 204, 212, 228, 276}. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 431–450, 1998     

The packing of pairs by quadruples

Let X be a finite set of size ν, further let λ be a positive integer and let σ(4,λν) denote the maximum number of quadruples such that each pair of elements of X is contained in at most λ of them. The value of σ(4, 1;ν) has been determined by Brouwer (1979) for all v4. The value of σ(4,λν) has been determined by Billington, Stanton and Stinson (1984) for all ν≡0 (mod 3) and λ>1. In this paper we complete the determination of σ(4,λν) for all ν4 and λ>1.     

On perfect pairs for quadruples in complemented modular lattices and concepts of perfect elements

Gel’fand and Ponomarev [11] introduced the concept of perfect elements and constructed such in the free modular lattice on 4 generators. We present an alternative construction of such elements u (linearly equivalent to theirs) and for each u a direct decomposition u, [`(u)]{\bar{u}} of the generating quadruple within the free complemented modular lattice on 4 generators; u, [`(u)]{\bar{u}} are said to form a perfect pair. This builds on [17] and fills a gap left there. We also discuss various notions of perfect elements and relate them to preprojective and preinjective representations.     

On turning points

For a family {T + N_λ: λ ? [a, b]} of semilinear operators T + N_λ in L₂(Ω) the solution set {(λ, u_λ) ? J × D(T): Tu_λ + N_λu_λ = h} is investigated with respect to turning points. By Ljapunov-Schmidt-reduction and calculation of the derivatives of the bifurcation equation a class of turning points is characterized by properties of these derivatives.     

Characterization of nearly Schroeder-Bernstein quadruples for Banach spaces

Let X and Y be Banach spaces such that each of them is isomorphic to a complemented subspace of the other. In 1996, W. T. Gowers solved the Schroeder-Bernstein problem for Banach spaces by showing that X is not necessarily isomorphic to Y . Let (p, q, r, s) be a quadruple in with p + q  ≥  2 and r + s ≥  2. Suppose that for every pair of Banach spaces X and Y isomorphic to complemented subspaces of each other and satisfying the following Decomposition Scheme

On Strebel points

Any Beltrami coefficient μ on a hyperbolic Riemann surface X of infinite type represents a point [μ] _T in the Teichmüller space T(X) and a point [μ] _B in the tangent space of T(X) at the base point as well. The paper deals with the problem of determining whether that [μ] _T is a Strebel point is equivalent to that [μ] _B is an infinitesimal Strebel point.     

Permutation invariant properties of primitive cubic quadruples

Based on a specific quadratic Hopf map between the Euclidean spaces of dimension four and three that is associated with Euler’s complete rational parameterization of the four cubes equation, we study the permutation invariant properties of the primitive integer cubic quadruples that solve this equation. Fixing the coordinate with maximum height and taking it positive, our main result describes the six positive primitive triples that leave it invariant under the inverted cubic map to this Hopf map and permute the remaining integer coordinates. The obtained invariant primitive triples are ordered in the so-called integer triple ordering, so that the minimum triple with respect to this ordering determines each primitive cubic quadruple uniquely. Implications for the counting and enumeration of all primitive cubic quadruples are mentioned. A list of all primitive cubic quadruples with positive maximum height below 100 and their minimum invariant triples is given. The relationship with the famous Taxicab and Cabtaxi numbers is also explained.     

The Weierstrass gap sets for quadruples II

Let M be a compact Riemann surface of genus g. Let P _i (i = 1, ..., 4) be 4 distinct points on M. We denote G(P ₁, P ₂, P ₃, P ₄) the Weierstrass gap set. In this paper, we prove that, for large g, the upper bound of #G(P ₁, P ₂, P ₃, P ₄) is attained if and only if M is hyperelliptic and |2P _i | = g ₂¹.