The classification of spreads in PG(3, q) admitting linear groups of order q(q+1), I. odd order

A classification is given of all spreads in PG(3, q), q = p^r, p odd, whose associated translation planes admit linear collineation groups of order q(q +1) such that a Sylow p-subgroup fixes a line and acts non-trivially on it.The authors are indebted to T. Penttila for pointing out the special examples of conical flock translation planes of order q² that admit groups of order q(q+1), when q = 23 or 47.     

Classification of triply-retractive semifields of order q 4

A complete classification of the semifield spreads of order q⁴ that triply-retract to mixed partitions of PG (3, q²) is given.     

Convexity of the inverse and Moore-Penrose inverse

Convexity properties of the inverse of positive definite matrices and the Moore-Penrose inverse of nonnegative definite matrices with respect to the partial ordering induced by nonnegative definiteness are studied. For the positive definite case null-space characterizations are derived, and lead naturally to a concept of strong convexity of a matrix function, extending the conventional concept of strict convexity. The positive definite results are shown to allow for a unified analysis of problems in reproducing kernel Hilbert space theory and inequalities involving matrix means. The main results comprise a detailed study of the convexity properties of the Moore-Penrose inverse, providing extensions and generalizations of all the earlier work in this area.     

Integral Representation of Some Functions Related to the Gamma Function

We prove that the functions and are Stieltjes transforms.     

Non-self-adjoint Dirac-type Systems and Related Nonlinear Equations: Wave Functions, Solutions, and Explicit Formulas

A version of the B?cklund-Darboux transformation, where Darboux matrix takes the form of the transfer matrix function from the system theory, for the non-self-adjoint Dirac-type system is considered. Related nonlinear Schr?dinger equations (coupled and multi-component), self-induced transparency equation, and non-Abelian sine-Gordon equation are treated. Explicit formulas for the wave functions and solutions are obtained.     

On bounds for the range of ordered variates II

Letx ₁, ,x _n be real numbers with ₁ ⁿ x _j =0, |x ₁ ||x ₂ ||x _n |, and ₁ ⁿ f(|x _i |)=A>0, wheref is a continuous, strictly increasing function on [0, ) withf(0)=0. Using a generalized Chebycheff inequality (or directly) it is easy to see that an upper bound for |x _m | isf ^–1 (A/(n–m+1)). If (n–m+1) is even, this bound is best possible, but not otherwise. Best upper bounds are obtained in case (n–m+1) is odd provided either (i)f is strictly convex on [0, ), or (ii)f is strictly concave on [0, ). Explicit best bounds are given as examples of (i) and (ii), namely the casesf(x)=x ^p forp>1 and 0<p<1 respectively.     

Multiplied Configurations, Series Induced by Correlations

We present a method of constructing configurations consisting in multiplying self-dual partial linear spaces involving their (fixed) correlations. Elementary (axiomatic) properties preserved under this construction are examined, and automorphism groups of the resulting configurations are determined.     

On the functional equationf(x) g(y) = p(x + y) q(x/y)

Aequationes mathematicae -     

Subpermanents of doubly stochastic matrices

It is shown that if all subpermaneats of order k of an n × n doubly stochastic matrix are equal for some k ≤ n - 2, then all the entries of the matrix must be equal to 1/n.     

Combinatorial Veronese Structures, Their Geometry, and Problems of Embeddability

Finite generalized Veronese spaces of special type (associated with single-line geometry) are studied, and problems concerning embeddability of the resulting configurations are discussed. Received: October 17, 2005. Revised: February 13, 2006.     

A Geometric Construction for Some Ovoids of the Hermitian Surface

Multiple derivation of the classical ovoid of the Hermitian surface of is a well known, powerful method for constructing large families of non classical ovoids of . In this paper, we shall provide a geometric costruction of a family of ovoids amenable to multiple derivation.