Walter Schempp

THE BRAIN AS A CONSCIOUS SYSTEM

A mathematically specified model of the brain/mind working by the physically realisable processes of quantum holography is proposed, in which mental events autonomously cause neural events. These are made manifest by adaptive resonance so as to elicit the stream of consciousness of which we are all aware. This takes place against a background of unconscious activity, for example the taxonomization and storage of sensory experience and mental activity in the form of a distributed, paged, holographic memory. The very considerable benefits of this form of memory and the associated image and signal processing by adaptive filtering are explained. It is shown that such a model exhibits many of the basic features of the actual morphology and dynamics of human brains including the possibility of language and logic as enhancements to the primary capabilities of perception and cognition. It confirms the nature of our personal experience as having a conscious mental self or ghost in the machine, in accordance with t...

On the Charshiladze-Lozinski theorem for compact topological groups and homogeneous spaces

This paper is concerned with an extension of the Charshiladze-Lozinski theorem to compact (not necessarily abelian) topological groups G and symmetric compact homogeneous spaces G/H. The proof is based on a generalized Marcinkiewicz — Berman formula. As an application, some divergence theorems for expansions of continuous resp. integrable complex — valued functions on Euclidean spheres and projective spaces in series of polynomial functions on these spaces are established.

Classical and quantum Heisenberg groups, their representations and applications

This paper is a survey on classical Heisenberg groups and algebras, q-deformed Heisenberg algebras, q-oscillator algebras, their representations and applications. Describing them, we tried, for the readers convenience, to explain where the q-deformed case is close to the classical one, and where there are principal differences. Different realizations of classical Heisenberg groups, their geometrical aspects, and their representations are given. Moreover, relations of Heisenberg groups to other linear groups are described. Intertwining operators for different (Schrödinger, Fock, compact) realizations of unitary irreducible representations of Heisenberg groups are given in explicit form. Classification of irreducible representations and representations of the q-oscillator algebra is derived for the cases when q is not a root of unity and when q is a root of unity. The Fock representation of the q-oscillator algebra is studied in detail. In particular, q-coherent states are described. Spectral properties of some operators of the Fock representations of q-oscillator algebras are given. Some of applications of Heisenberg groups and algebras, q-Heisenberg algebras and q-oscillator algebras are briefly described.

The Moyal representation of quantum mechanics and special function theory

It is shown that the phase-space formulation of quantum mechanics is a rich source of special function identities. The Moyal formalism is reviewed for two phase spaces: the real plane and the sphere; and this is used to derive identities for Airy, Laguerre, Kummer, and theta functions and for SU(2) rotation elements, several of which are new.

Sub-Riemannian geometry and clinical magnetic resonance tomography

Mensch, streckh deine Vernunfft hieher, diese dinge zu begreiffen! Johann Keppler (1571–1630) There is nothing that nuclear spins will not do for you, as long as you treat them as human beings. Erwin Louis Hahn (1949) Ein maschinelles ‘agencement’ ist den Schichten zugewandt, reinen Intensitaten, die sie zirkulieren last um die Selektion der ‘Konsistenzebene’ zu sichern, und der sich die Subjekte zuordnen, welchen sie einen Namen nur als Spur einer Intensitat last. Gilles Deleuze und Felix Guattari (1988) The sense we make of the world is governed by our conceptual means. More specifically, sense is made of nature by projecting a conceptual structure onto observed events. This is accomplished by means of a semantic filter, which transforms raw data into semantically significant events. George L. Farre (1997) Phase coherent wavelets form a unified basis of the multichannel reconstruction analysis–synthesis filter bank, implemented by high-resolution synthetic aperture radar (SAR) imaging and clinical MRI. The construction of unitary bank filters is performed by the Kepplerian stratification technique. It allows for the cross-sectional quadrature filtering of phase histories in contiguous local frequency encoding subband multichannels relative to the rotating coordinate frame of quadrature reference. The Kepplerian quadrature detection strategy, which is based on the unitary filter bank construction takes place in symplectic affine planes of incidence immersed into the three-dimensional compact super-encoding projective space. The planes are implemented in the quadrature format of a phase-splitting network of Fourier analysis of the stratified Heisenberg nilpotent Lie group G. The action of the basic Lie group G of quantum physics under its sub-Riemannian geometry admits a matrix coordinatization by transvections. In terms of projective geometry, the longitudinal dilations act as a spectral transform on the transvections of the G-action. The tomographic slices, frequency selected by the MRI scanner system, are identified with the projectively immersed, symplectic affine leaves ν (ν=0) of the canonical coadjoint orbit foliation of G, on which the continuous affine wavelet transform lifts to a transversal spectral transform. The paper points out that the key point of ensemble quantum computation by Fourier transform MRI is the synergy of the various wavelets, and the coexistence of their geometric substrates recorded by the bi-infinite stratigraphic time line. The paper leads from Keppler to Heisenberg and emphasizes the tight control which Lie groups exercise over the non-invasive MRI modality via geometric quantization. The fascinating aspects of electronic engineering concerned with the calculation of the Weyl symbol are also considered. Copyright

Radar Ambiguity Functions, Nilpotent Harmonic Analysis, and Holomorphic Theta Series

As is well known, radar (=abbreviation of RAdio Detection And Ranging) systems are a device for discovering distant objects that are stationary or moving such as ships, aeroplanes, and satellites. Besides the detection of the presence of a remote target, the purpose of a radar system is basically to extract information of interest (such as range, relative velocity, etc.) about the target. The radar transmitter generates electromagnetic energy of a few centimeters’ wavelength in the form of pulses of large amplitude and brief duration which are emitted periodically through an antenna that produces a narrow beam of radiation. Any object located in the path of the propagating beam scatters the radiation in all directions and a small portion of the scattered radiation excites the receiving antenna. It can be achieved by means of modern electronical equipments that the radar system uses a common antenna for both transmission and reception: In an elementary form of a radar system a duplexer enables the radar antenna to operate in the transmission mode as well as in the reception mode. The reflected signal energy picked up by the radar antenna (operating in the reception mode) is led to a receiver, amplified, and then applied to the vertical deflection plates of a cathode-ray oscilloscope to detect the presence of the radar target and estimate its parameters.

Heisenberg groups: a unifying structure of signal theory, holography and quantum information theory

Vector fields in three-space admit bundles of internal variables such as a Heisenberg algebra bundle. Information transmission along field lines of vector fields is described by a wave linked to the Schrödinger representation in the realm of time-frequency analysis. The preservation of local information causes geometric optics and a quantization scheme. A natural circle bundle models quantum information visualized by holographic methods. Features of this setting are applied to magnetic resonance imaging.

Vector Fields in Three-Space, Natural Internal Degrees of Freedom, Signal Transmission and Quantization

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Radar Ambiguity Functions of Positive Type

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Radar Ambiguity Functions and the Linear Schrödinger Representation

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