On the maximality of linear codes

We show that if a linear code admits an extension, then it necessarily admits a linear extension. There are many linear codes that are known to admit no linear extensions. Our result implies that these codes are in fact maximal. We are able to characterize maximal linear (n, k, d) _q -codes as complete (weighted) (n, n − d)-arcs in PG(k − 1, q). At the same time our results sharply limit the possibilities for constructing long non-linear codes. The central ideas to our approach are the Bruen-Silverman model of linear codes, and some well known results on the theory of directions determined by affine point-sets in PG(k, q).      

On maximum principles for a class of nonlinear second-order elliptic equations

In a recent paper [3] the authors derived maximum principles which involved $\text{u(x)}\text{and}\text{q = ¦}\text{grad}\text{u¦}$, where u(x) is a classical solution of an alliptic differential equation of the form ($\text{g(q}{}^2\text{)u}{}_{\mathrm{,i}}\text{)}{}_{\mathrm{,i}}\text{+ ?(u) ?(q}{}^2\text{) = 0}$. In this paper these results are extended to the more general case in which $\text{g = g(u, q}{}^2\text{)}\text{and}\text{?(u) ?(q}{}^2\text{)}$ is replaced by h(u, q²).     

On the maximality of a set of mutually orthogonal Sudoku Latin Squares

The maximum number of mutually orthogonal Sudoku Latin squares (MOSLS) of order \(n=m^2\) is \(n-m\). In this paper, we construct for \(n=q^2\), q a prime power, a set of \(q^2-q-1\) MOSLS of order \(q^2\) that cannot be extended to a set of \(q^2-q\) MOSLS. This contrasts to the theory of ordinary Latin squares of order n, where each set of \(n-2\) mutually orthogonal Latin Squares (MOLS) can be extended to a set of \(n-1\) MOLS (which is best possible). For this proof, we construct a particular maximal partial spread of size \(q^2-q+1\) in \(\mathrm {PG}(3,q)\) and use a connection between Sudoku Latin squares and projective geometry, established by Bailey, Cameron and Connelly.     

Variational principles and variational—iterative principles for a class of problems

Complementary variational principles are developed for linear equations in a function space and a variational-iterative scheme, based on these principles, is introduced for non-linear equations.     

Local maximality of hyperbolic sets

Two properties of a hyperbolic set F are discussed: its local maximality and the property that, in any neighborhood U ⊃ F, there exists a locally maximal set F′ that contains F (we suggest calling the latter property local premaximality). Although both these properties of the set F are related to the behavior of trajectories outside F, it turns out that, in the class of hyperbolic sets, the presence or absence of these properties is determined by the interior dynamics on F.     

Extremum principles for a general class of saddle functionals

The usual notion of a saddle functional in the calculus of variations assumes a vex/concave structure over the product space of two inner product spaces. Here the ideas extended to include some convexity in both spaces whilst still retaining an overall saddle property. Dual extremum principles are established for these functionals. Examples include periodic solutions of Duffing's equation, an iterative scheme and a pair of simultaneous partial differential equations which arise in magnetohydrodynamics.     

Variational principles for a class of boundary value problems

A variational formulation is developed for boundary value problems described by operator equations ( + ^*)h=w(h) in some region V, subject to b(h) = 0 on the boundary of V.     

Variational principles for a class of coupled linear equations

This paper presents variational and extremum principles for pairs of coupled linear equations. The results are illustrated by several examples arising in mathematical physics.     

The maximality of the core model

Our main results are: 1) every countably certified extender that coheres with the core model is on the extender sequence of , 2) computes successors of weakly compact cardinals correctly, 3) every model on the maximal 1-small construction is an iterate of , 4) (joint with W. J. Mitchell) is universal for mice of height whenever , 5) if there is a such that is either a singular countably closed cardinal or a weakly compact cardinal, and fails, then there are inner models with Woodin cardinals, and 6) an -Erdös cardinal suffices to develop the basic theory of .

     

Groups with a maximality condition for some non-normal subgroups

We describe (generalized) soluble-by-finite groups in which the set of non-normal subgroups which are not finitely generated satisfies the maximal condition.     

Effect algebras with the maximality property

The maximality property was introduced in orthomodular posets as a common generalization of orthomodular lattices and orthocomplete orthomodular posets. We show that various conditions used in the theory of effect algebras are stronger than the maximality property, clear up the connections between them and show some consequences of these conditions. In particular, we prove that a Jauch–Piron effect algebra with a countable unital set of states is an orthomodular lattice and that a unital set of Jauch–Piron states on an effect algebra with the maximality property is strongly order determining.