Quasideterminant solutions of a non-Abelian Hirota-Miwa equation

A non-Abelian version of the Hirota-Miwa equation is considered. In an earlier paper [Nimmo (2006) J. Phys. A: Math. Gen. \textbf{39}, 5053-5065] it was shown how solutions expressed as quasideterminants could be constructed for this system by means of Darboux transformations. In this paper we discuss these solutions from a different perspective and show that the solutions are quasi-Pl\"{u}cker coordinates and that the non-Abelian Hirota-Miwa equation may be written as a quasi-Pl\"{u}cker relation. The special case of the matrix Hirota-Miwa equation is also considered using a more traditional, bilinear approach and the techniques are compared

On pattern structures of the N-soliton solution of the discrete KP equation over a finite field

The existence and properties of coherent pattern in the multisoliton solutions of the dKP equation over a finite field is investigated. To that end, starting with an algebro-geometric construction over a finite field, we derive a "travelling wave" formula for $N$-soliton solutions in a finite field. However, despite it having a form similar to its analogue in the complex field case, the finite field solutions produce patterns essentially different from those of classical interacting solitons.Comment: 12 pages, 3 figure

On a direct approach to quasideterminant solutions of a noncommutative KP equation

A noncommutative version of the KP equation and two families of its solutions expressed as quasideterminants are discussed. The origin of these solutions is explained by means of Darboux and binary Darboux transformations. Additionally, it is shown that these solutions may also be verified directly. This approach is reminiscent of the wronskian technique used for the Hirota bilinear form of the regular, commutative KP equation but, in the noncommutative case, no bilinearising transformation is available.Comment: 11 page

Tonotopic representation of loudness in the human cortex

A prominent feature of the auditory system is that neurons show tuning to audio frequency; each neuron has a characteristic frequency (CF) to which it is most sensitive. Furthermore, there is an orderly mapping of CF to position, which is called tonotopic organization and which is observed at many levels of the auditory system. In a previous study (Thwaites et al., 2016) we examined cortical entrainment to two auditory transforms predicted by a model of loudness, instantaneous loudness and short-term loudness, using speech as the input signal. The model is based on the assumption that neural activity is combined across CFs (i.e. across frequency channels) before the transform to short-term loudness. However, it is also possible that short-term loudness is determined on a channel-specific basis. Here we tested these possibilities by assessing neural entrainment to the overall and channel-specific instantaneous loudness and the overall and channel-specific short-term loudness. The results showed entrainment to channel-specific instantaneous loudness at latencies of 45 and 100 ms (bilaterally, in and around Heschl's gyrus). There was entrainment to overall instantaneous loudness at 165 ms in dorso-lateral sulcus (DLS). Entrainment to overall short-term loudness occurred primarily at 275 ms, bilaterally in DLS and superior temporal sulcus. There was only weak evidence for entrainment to channel-specific short-term loudness.This work was supported by an ERC Advanced Grant (230570, ‘Neurolex’) to WMW, by MRC Cognition and Brain Sciences Unit (CBU) funding to WMW (U.1055.04.002.00001.01), and by EPSRC grant RG78536 to JS and BM

Matrix solutions of a noncommutative KP equation and a noncommutative mKP equation

Matrix solutions of a noncommutative KP and a noncommutative mKP equation which can be expressed as quasideterminants are discussed. In particular, we investigate interaction properties of two-soliton solutions.Comment: 2 figure

The difficult legacy of Turing’s wager

Describing the human brain in mathematical terms is an important ambition of neuroscience research, yet the challenges remain considerable. It was Alan Turing, writing in 1950, who first sought to demonstrate how time-consuming such an undertaking would be. Through analogy to the computer program, Turing argued that arriving at a complete mathematical description of the mind would take well over a thousand years. In this opinion piece, we argue that — despite seventy years of progress in the field — his arguments remain both prescient and persuasive