Exotic complex Hadamard matrices, and their equivalence

In this paper we use a design theoretical approach to construct new, previously unknown complex Hadamard matrices. Our methods generalize and extend the earlier results of de la Harpe--Jones and Munemasa--Watatani and offer a theoretical explanation for the existence of some sporadic examples of complex Hadamard matrices in the existing literature. As it is increasingly difficult to distinguish inequivalent matrices from each other, we propose a new invariant, the fingerprint of complex Hadamard matrices. As a side result, we refute a conjecture of Koukouvinos et al. on (n-8)x(n-8) minors of real Hadamard matrices.Comment: 10 pages. To appear in Cryptography and Communications: Discrete Structures, Boolean Functions and Sequence

Reconstructing Projective Planes from Semibiplanes

Abstract. From a projective plane Π with a homology τ of order 2, one obtains an incidence system having as points and blocks the 〈τ〉-orbits of length 2 on the points and lines of Π, and with incidence inherited from Π. The resulting structure, denoted by Π/τ, is an example of a homology semibiplane. We have shown that a Desarguesian projective plane of odd prime order is uniquely reconstructible from its homology semibiplane (although such a reconstruction is not in general unique for other planes). This is one step towards classifying projective planes of prime order which admit a collineation of order 2. More generally we reduce the problem of ‘lifting ’ semibiplanes to projective planes, to an equivalent (but better codified) problem in linear algebra. Conceivably this technique may produce new projective planes from the semibiplanes of known planes

Emerging technologies for improved deep brain stimulation

Deep brain stimulation (DBS) is an effective treatment for common movement disorders and has been used to modulate neural activity through delivery of electrical stimulation to key brain structures. The long-term efficacy of stimulation in treating disorders, such as Parkinson’s disease and essential tremor, has encouraged its application to a wide range of neurological and psychiatric conditions. Nevertheless, adoption of DBS remains limited, even in Parkinson’s disease. Recent failed clinical trials of DBS in major depression, and modest treatment outcomes in dementia and epilepsy, are spurring further development. These improvements focus on interaction with disease circuits through complementary, spatially and temporally specific approaches. Spatial specificity is promoted by the use of segmented electrodes and field steering, and temporal specificity involves the delivery of patterned stimulation, mostly controlled through disease-related feedback. Underpinning these developments are new insights into brain structure–function relationships and aberrant circuit dynamics, including new methods with which to assess and refine the clinical effects of stimulation