The revised Robertson's test theory of special relativity (SR) has been constructed upon a family of sets of passive coordinate transformations in flat space-time [J. G. Vargas and D. G. Torr,Found. Phys., 16, 1089 (1986)]. In the same paper, it has also been shown that the boosts depend in general on the velocities of the two frames involved and not only on their relative velocity. The only exception to this is SR, if one has previously used an appropriate constraint to (...) remove the other relativities—like Galilean relativity—from the family.In this paper we look at these coordinate transformations in the only way there is to do so, namely as transformations in a seven-dimensional “Cartan Space” (Cartan first considered this in his dealings with Newtonian kinematics). In this space, the boosts only depend on the relative velocity of the frames. The passive coordinate transformations in each set are shown to have a nonlinear group structure isomorphic to that of the Poincaré group.The existence of a preferred frame, except in SR, makes the active transformations inequivalent to the passive ones. It is shown that the composite active-passive transformations act on a ten-dimensional space and that each member set of the family also has a group structure. As a result, one ends up with a family of mutually isomorphic 9-parameter (homogeneous) supergroups and a family of mutually isomorphic (9 + 4)-parameter (inhomogeneous) supergroups. The presence of extra parameters could be looked upon as “internal” degrees of freedom, which are, however, an offshoot of the Robertson space-time. (shrink)

We define a notion of genericity for genericity subgroups of groups interpretable in a simple theory. and show that a type generic for such a group is generic for the minimal hyperdefinable supergroup (the definable hull). In particular, at least one generic type of the definable hull is finitely satisfiable in the original subgroup. If the subgroup is a subfield, then the additive and the multiplicative definable hull both have bounded index in the smallest hyperdefinable superfield.

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Embodied Knowledge in Ensemble Performance by J. Murphy McCalebEric C. Melley, D.M.A.J. Murphy McCaleb, Embodied Knowledge in Ensemble Performance (Surrey, England: Ashgate, 2014)J. Murphy McCaleb’s Embodied Knowledge in Ensemble Performance explores how musicians interact and share information while performing, specifically within unconducted chamber ensembles. The book is a direct outgrowth of the author’s doctoral dissertation and follows a similar format. Beginning with a presentation of four essential research (...) questions, McCaleb continues with an overview and justification for his methodology, followed by case studies, conclusions, and foundations for further research. The four questions McCaleb seeks to answer with this volume are: How do musicians interact and share information with each other while performing? What is the nature of the information being shared in ensemble performance? To what extent does the musical content being performed affect the ways it has to be physically created by musicians? How does the physical relationship between the performers and their instruments relate to communicative and interactive processes of ensemble performance? After an explanation of methodology, McCaleb devotes a chapter to each of the four questions. The substantial and excellent review of literature is distributed throughout the comprising six chapters, providing a natural connection between case studies, previous research, and resultant conclusions. While the author suggests the book [End Page 103] has value for both researchers and practitioners, the most prominent result of the discourse provides a new paradigm with which to study ensemble interaction, but contains few practical suggestions for performers, leaning heavily on previous research. McCaleb’s work will most directly benefit other researchers and music education theorists.The volume spans 140 pages, including an index and bibliography, as well as an accompanying DVD. The DVD includes video examples of rehearsals and performances for two ensembles that served as subjects of the research case studies: the Boult String Quartet, the premiere student quartet at the Birmingham Conservatoire (where the author was completing his doctorate), and The Supergroup, an ensemble that performs free group improvisations. The author is himself a member of The Supergroup and serves as “a reflective practitioner, assuming the roles of researcher and musician” (p.16). While his research methodology is pluralistic (though decidedly qualitative in nature), the employment of action research, wherein the “people being studied become part of the knowledge creation process,” is a prominent element in his method (p. 13). This is one area where McCaleb’s perspective is both powerful and open to debate. One of the central and strongest premises of McCaleb’s work is that music is a singular form of procedural knowledge. Because the nature of procedural knowledge at some level defies linguistic translation, the perspective of performing musicians is a critical element to any phenomenological study of musical performance. The author acknowledges that this raises questions of subjectivity and works to incorporate empirical methods such as “observation, interviews, literature review, and case studies” to counter the more subjective methodology (p.15). McCaleb’s multifaceted approach to research is for the most part highly effective, but at times the anecdotal tone of his first-hand descriptions of musical experiences can soften the force of his premise.Based on the concept that music is a form of procedural knowledge, McCaleb posits that communication among ensemble musicians is first and foremost musical. In order to illustrate this, the author explores other common models used to study interaction between musicians. In this review of other research studies, McCaleb is most effective in pointing out the inadequacies of applying other models to musical ensemble interaction. The linguistic model, which seeks to correlate physical gestures with speech patterns, remains inadequate to study communication between instrumental musicians. Previous research investigating the gestural model, while focusing on instrumental music, is preoccupied with emotional content, primarily between performer and audience, and does not approach the question of what and how communication occurs between performing musicians (pp. 27–28). Based on the inadequacies of other research [End Page 104] models, McCaleb seeks to establish a new paradigm of interaction entirely independent of social models and based solely in musical performance (p. 7).This points to an important area of research that McCaleb largely neglects: the neuroscience... (shrink)

The various physical aspects of the general relativistic principles of covariance and strong equivalence are discussed, and their mathematical formulations are analyzed. All these aspects are shown to be present in classical general relativity, although no contemporary formulation of canonical or covariant quantum gravity has succeeded to incorporate them all. This has, in part, motivated the recent introduction of a geometro-stochastic framework for quantum general relativity, in which the classical frame bundles that underlie the formulation of parallel transport in classical (...) general relativity are replaced by quantum frame bundles. It is shown that quantum frames can take over the role played by complete sets of observables in conventional quantum theory, so that they can mediate the natural transference of the general covariance and the strong equivalence principles from the classical to the quantum general relativistic regime. This results in a geometrostochastic mode of quantum propagation in general relativistic quantum bundles, which is mathematically implemented by path integration methods based on parallel transport along horizontal lifts of geodesics for the vacuum expectation values of a quantum gravitational field in a quantum spacetime supermanifold. The covariance features of this field are embedded in a quantum gravitational supergroup, which incorporates Poincaré as well as diffeomorphism invariance, and resolves the issue of time in quantum gravity. (shrink)

Assume T is unidimensional, 1-based and every minimal type in T is locally finite. If H is an Λ -definable irreducible group, we find an irreducible supergroup G of H in acleq such that any connected subgroup of Gn, n < ω, is the connected component of a subgroup linearly defined over the ring End*. In some cases we can take G = H.

Relativistic geometrical action for a quantum particle in the superspace is analyzed from theoretical group point of view. To this end an alternative technique of quantization outlined by the authors in a previous work and that is based in the correct interpretation of the square root Hamiltonian, is used. The obtained spectrum of physical states and the Fock construction consist of Squeezed States which correspond to the representations with the lowest weights $\lambda=\frac{1}{4}$ and $\lambda=\frac{3}{4}$ with four possible (non-trivial) fractional representations (...) for the group decomposition of the spin structure. From the theory of semigroups the analytical representation of the radical operator in the superspace is constructed, the conserved currents are computed and a new relativistic wave equation is proposed and explicitly solved for the time-dependent case. The relation with the Relativistic Schrödinger equation and the Time-dependent Harmonic Oscillator is analyzed and discussed. (shrink)