Mutual definability does not imply definitional equivalence,a simple example

We give two theories, Th₁ and Th₂, which are explicitly definable over each other (i.e. the relation symbols of one theory are explicitly definable in the other, and vice versa), but are not definitionally equivalent. The languages of the two theories are disjoint. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On evaluating multivariate polynomials over finite fields

Some evaluation methods of multivariate polynomials over finite fields are described and their multiplicative complexity is discussed.     

3-Manifold group and finite decomposition complexity

In this paper, we prove that the fundamental group of an orientable compact 3-manifold has finite decomposition complexity if Thurston's Hyperbolization Conjecture is true.     

Hensel lifting and bivariate polynomial factorisation over finite fields

This paper presents an average time analysis of a Hensel lifting based factorisation algorithm for bivariate polynomials over finite fields. It is shown that the average running time is almost linear in the input size. This explains why the Hensel lifting technique is fast in practice for most polynomials.

     

Computing irreducible representations of supersolvable groups over small finite fields

We present an algorithm to compute a full set of irreducible representations of a supersolvable group over a finite field , , which is not assumed to be a splitting field of . The main subroutines of our algorithm are a modification of the algorithm of Baum and Clausen (Math. Comp. 63 (1994), 351-359) to obtain information on algebraically conjugate representations, and an effective version of Speiser's generalization of Hilbert's Theorem 90 stating that vanishes for all .

     

Modelling of laminate structures in elasto-plasticity with a mixed finite element

The main advantage of the mixed finite element (displacements-stresses) is that, because of its continuity, it gives a good approximate stress field, without needing a high degree of interpolation in displacements. This degree of approximation is essential for elastic-plastic computations. However, the total continuity of stresses is too strong in a laminate structure along the interfaces. We show a method of achieving the correct level of continuity without losing the advantage of a good approximation. Some examples of laminate plane structures with plastic areas along the interfaces are given.     

Symbolic analysis of finite words: the complexity function

We present several properties of the complexity function of finite words, the function counting the number of different factors in a word, for each length. To establish a first set of properties, we use the de Bruijn graphs and the suffix tree representations of a word. This allows us to show some inequalities that control the variation as well as the maximal value of the complexity function. Motivated by the applications, we discuss the change of the complexity function when sliding or increasing the size of a window laid down on a sequence to be analysed.     

Substandard models of finite set theory

A survey of the isomorphic submodels of V_ω, the set of hereditarily finite sets. In the usual language of set theory, V_ω has 2^?0 isomorphic submodels. But other set‐theoretic languages give different systems of submodels. For example, the language of adjunction allows only countably many isomorphic submodels of V_ω (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal curves of low genus over finite fields

We investigate maximal and minimal curves of genus 4 and 5 over finite fields with discriminant −11 and −19. As a result the Hasse–Weil–Serre bound is improved.     

Modal and guarded characterisation theorems over finite transition systems

We explore the finite model theory of the characterisation theorems for modal and guarded fragments of first-order logic over transition systems and relational structures of width two. A new construction of locally acyclic bisimilar covers provides a useful analogue of the well known tree-like unravellings that can be used for the purposes of finite model theory. Together with various other finitary bisimulation respecting model transformations, and Ehrenfeucht–Fraïssé game arguments, these covers allow us to upgrade finite approximations for full bisimulation equivalence towards approximations for elementary equivalence. These techniques are used to prove several ramifications of the van Benthem–Rosen characterisation theorem of basic modal logic for refinements of ordinary bisimulation equivalence, both in the sense of classical and of finite model theory.     

Determinants of matrices over commutative finite principal ideal rings

In this paper, the determinants of $n\times n$ matrices over commutative finite chain rings and over commutative finite principal ideal rings are studied. The number of $n\times n$ matrices over a commutative finite chain ring R of a fixed determinant a is determined for all $a\in R$ and positive integers n. Using the fact that every commutative finite principal ideal ring is a product of commutative finite chain rings, the number of $n\times n$ matrices of a fixed determinant over a commutative finite principal ideal ring is shown to be multiplicative, and hence, it can be determined. These results generalize the case of matrices over the ring of integers modulo m.     

Arithmetic of divisibility in finite models

We prove that the finite‐model version of arithmetic with the divisibility relation is undecidable (more precisely, it has Π⁰₁‐complete set of theorems). Additionally we prove FM‐representability theorem for this class of finite models. This means that a relation R on natural numbers can be described correctly on each input on almost all finite divisibility models if and only if R is of degree ≤ 0 ′. We obtain these results by interpreting addition and multiplication on initial segments of finite models with divisibility only. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Subquadratic-time factoring of polynomials over finite fields

New probabilistic algorithms are presented for factoring univariate polynomials over finite fields. The algorithms factor a polynomial of degree over a finite field of constant cardinality in time . Previous algorithms required time . The new algorithms rely on fast matrix multiplication techniques. More generally, to factor a polynomial of degree over the finite field with elements, the algorithms use arithmetic operations in .

The new ``baby step/giant step' techniques used in our algorithms also yield new fast practical algorithms at super-quadratic asymptotic running time, and subquadratic-time methods for manipulating normal bases of finite fields.

     

A Laplace transform technique for the analytical solution of a diffusion- convection equation over a finite domain

A Laplace transform technique has been utilized to obtain two different analytic solutions to a single diffusion-convection equation over a finite domain. One analytic solution is continuous at both ends of the domain of interest, while the other solution is discontinuous at the origin. This difference in the two solutions is explained. An application of the Laplace transform technique to a more complex system of equations, on a finite domain, is noted and an error apparent in a previous paper is corrected.     

The independence of two randomness properties of sequences over finite fields

We prove the logical independence of a complexity-theoretic and a statistical randomness property of sequences over a finite field. The two properties relate to the linear complexity profile and to the ∞$\infty$-distribution of sequences, respectively. The proofs are given by constructing counterexamples to the presumed logical implications between these two properties.