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.     

Primitive semifields of order $$2^{4e}$$

We prove that all sufficient large finite semifields of center \({\mathbb {F}}_{2^e}\) and order \(2^{4e}\) are right and left primitive.     

Fractional dimensions in semifields of odd order

A finite semifield D is considered a fractional dimensional semifield if it contains a subsemifield E such that λ := log_|E||D| is not an integer. We develop spread-theoretic tools to determine when finite planes admit coordinatization by fractional semifields, and to find such semifields when they exist. We use our results to show that such semifields exist for prime powers 3 ⁿ whenever n is an odd integer divisible by 5 or 7.     

A class of unitals of order q which can be embedded in two different planes of order q2

By deriving the desarguesian plane of order q² for every prime power q a unital of order q is constructed which can be embedded in both the Hall plane and the dual of the Hall plane of order q² which are non-isomorphic projective planes. The representation of translation planes in the fourdimensional projective space of J. André and F. Buekenhouts construction of unitals in these planes are used. It is shown that the full automorphism groups of these unitals are just the collineation groups inherited from the classical unitals.     

On the existence of semifields of prime power order admitting free automorphism groups

The purpose of this paper is to prove the existence of semifields of order q ⁴ for any odd prime power q = p^r, q > 3, admitting a free automorphism group isomorphic to Z ₂ × Z ₂.     

A new class of order types

Let φ₄ be the class of all order-types ϕ with the properties that every uncountable subtype of ϕ contains an uncountable well-ordering, but ϕ is not the union of countably many well-orderings. It is proved that φ₄ ≠ 0, and a way is found of associating stationary sets with most of the types in φ₄ which is useful for applications. A number of results concerning the structure and embeddability properties of φ₄ are obtained, including some consistency and independence results. One consequence is the independence of Jensen's combinatorial principle □_ω1.     

A structure theory for two-dimensional translation planes of order q 2 that admit collineation groups of order q 2

This paper is devoted to the study of translation planes of order q ² and kernel GF(q) that admit a collineation group of order q ² in the linear translation complement. We give a representation of this group by a suitable set of matrices depending on some functions over GF(q). Using this representation we obtain several results concerning the existence and the collineation group of the plane.     

A family of translation planes of order q3

In this note, some new class of translation planes of order q³, where q is an odd prime power with q 3,7, are constructed. The translation complement of any plane of this class has three orbits lengths 1, 1 and q³-1 on 1.     

A family of semifields in odd characteristic

We study a large family of semifields in odd characteristic, which contains the commutative Budaghyan–Helleseth semifields as well as semifields which are not isotopic to commutative semifields. Using a large group of autotopisms we obtain a complete classification result in certain parametric subcases.     

A new class of square order planes

Only a few classes of square order planes are known. These are generalized André planes (including Hall planes), flag transitive planes, Hering's planes and Walker's planes. A new class of planes of order 5^2r, where r is an odd natural number, is constructed and the translation complements of the corresponding planes have been determined. The translation complements divide the sets of distinguished points into 4 orbits of lengths 1, 1, 5^r ? 1, and 5^2r ? 5^r and it is of order 5^r(5^r ? 1)² if r ≠ 1. In the particular case of r = 1, the translation complement divides the set of distinguished points into two orbits of lengths 6 and 20 and it is of order 480.     

A class of first order factorization methods

A class of first order factorization methods for the solution of large, symmetric, sparse systems of equations is introduced. Asymptotic results for the computational complexity are developed, results from numerical experiments are presented and comparisons with other iterative and direct methods are carried out.     

A class of second order differential equations

We study the differential equationf″=N(f)f′ ² +M(f)f′+L(f), whereL, M, N are rational functions, and prove that if the differential equation has a transcendental meromorphic solutionf with order,p(f)>2, then the differential equation must be one of nine forms; and, moreover, we construct examples showing the existence of these nine forms with a transcendental meromorphic solution.     

A higher order p-adic class number formula

We generalize a formula of Leopoldt which relates the p-adic regulator modulo p of a real abelian extension of ? with the value of the relative Dedekind zeta function at s = 2 ? p. We use this generalization to give an alternative proof of the non-vanishing modulo p of this relative zeta function at the point s = 1 under a mild condition.