Equivalences of Zt×Z22-cocyclic Hadamard matrices

One of the most promising structural approaches to resolving the Hadamard Conjecture uses the family of cocyclic matrices over Zt × Z2 2. Two types of equivalence relations for classifying cocyclic matrices over Zt × Z2 2 have been found. Any cocyclic matrix equivalent by either of these relations to a Hadamard matrix will also be Hadamard. One type, based on algebraic relations between cocycles over any fi- nite group, has been known for some time. Recently, and independently, a second type, based on four geometric relations between diagrammatic visualisations of cocyclic matrices over Zt × Z2 2, has been found. Here we translate the algebraic equivalences to diagrammatic equivalences and show one of the diagrammatic equivalences cannot be obtained this way. This additional equivalence is shown to be the geometric translation of matrix transposition

A simple construction of complex equiangular lines

A set of vectors of equal norm in $\mathbb{C}^d$ represents equiangular lines if the magnitudes of the inner product of every pair of distinct vectors in the set are equal. The maximum size of such a set is $d^2$, and it is conjectured that sets of this maximum size exist in $\mathbb{C}^d$ for every $d \geq 2$. We describe a new construction for maximum-sized sets of equiangular lines, exposing a previously unrecognized connection with Hadamard matrices. The construction produces a maximum-sized set of equiangular lines in dimensions 2, 3 and 8.Comment: 11 pages; minor revisions and comments added in section 1 describing a link to previously known results; correction to Theorem 1 and updates to reference

Identification performance of evidential value estimation for ridge-based biometrics

Law enforcement agencies around the world use ridge-based biometrics, especially fingerprints, to fight crime. Fingermarks that are left at a crime scene and identified as potentially having evidential value (EV) in a court of law are recorded for further forensic analysis. Here, we test our evidential value algorithm (EVA) which uses image features trained on forensic expert decisions for 1428 fingermarks to produce an EV score for an image. First, we study the relationship between whether a fingermark is assessed as having EV, either by a human expert or by EVA, and its correct and confident identification by an automatic identification system. In particular, how often does an automatic system achieve identification when the mark is assessed as not having evidential value? We show that when the marks are captured by a mobile phone, correct and confident automatic matching occurs for 257 of the 1428. Of these, 236 were marked as having sufficient EV by experts and 242 by EVA thresholded on equal error rate. Second, we test four relatively challenging ridge-based biometric databases and show that EVA can be successfully applied to give an EV score to all images. Using EV score as an image quality value, we show that in all databases, thresholding on EV improves performance in closed set identification. Our results suggest an EVA application that filters fingermarks meeting a minimum EV score could aid forensic experts at the point of collection, or by flagging difficult latents objectively, or by pre-filtering specimens before submission to an AFIS

A quick test for nonisomorphism of one-relator groups

A sequence of integers is read off from the presentation of a finitely generated torsion-free one-relator group with nontrivial second integral homology, without recourse to group-theoretic manipulations. This test sequence is derived from the cup coproduct on the coring of the integral homology module of the group, and reflects information about the group's second lower central factor group. Test sequences differ only if the corresponding groups are nonisomorphic. The test process can be generalised to any one-relator group with nontrivial second integral homology

The diagonal comultiplication on homology

This paper describes the diagonal comultiplication (or cup coproduct) defined on integral homology modules of groups. Analysis of this coproduct should provide a new method of testing for non-isomorphism of groups which have isomorphic integral homology modules; here, the dimension two coproduct is applied to this problem. The first part (Section 2) is couched in terms of groupnets (Brandt groupoids) and shows two things: that there exists a cup product defined on the integral cohomology of any groupnet, extending that for groups, and that there exists a comultiplication defined on the integral homology of any group, natural up to dimension two, which gives the homology modules the structure of a commutative graded co-ring. In the second part (Sections 3 and 4), this diagonal comultiplication R is constructed to dimension two, and the information it carries about the lower central series of a group G is investigated. Modulo torsion in Hr(G; Z), Rz induces an abelian group homomorphism with cokernel GZ/G3, which distinguishes between large classes of groups, in particular the one-relator groups with non-trivial multiplicator, and the finitely-generated nilpotent groups of class two whose relators are all in the commutator subgroup

The mapping cylinder resolution for a groupnet diagram

No abstract availabl

The word problem and related results for graph product groups

A graph product is the fundamental group of a graph of groups. Amongst the simplest examples are HNN groups and free products with amalgamation. The torsion and conjugacy theorems are proved for any group presented as a graph product. The graphs over which some graph product has a solvable word problem are characterised. Conditions are then given for the solvability of the word and order problems and also for the extended word problem for cyclic subgroups of any graph product. These results generalise the known ones for HNN groups and free products with amalgamation