Abstract:
I apply the algebraic classification of self-adjoint endomorphisms of ${\bf R}^{2,2}$ provided by their Jordan canonical form to the Ricci curvature tensor of four-dimensional neutral manifolds and relate this classification to an algebraic classification of the Ricci curvature spinor. These results parallel similar results well known in four-dimensional Lorentzian geometry. The classification is summarized in Table 2 at the end of the paper.

Abstract:
By the use of complete orthonormal sets of nonrelativistic scalar orbitals introduced by the author in previous papers the new complete orthonormal basis sets for two- and four-component spinor wave functions, and Slater spinor orbitals useful in the quantum-mechanical description of the spin- 1/2 particles by the quasirelativistic and Dirac's relativistic equations are established in position, momentum and four-dimensional spaces. These function sets are expressed through the corresponding nonrelativistic orbitals. The analytical formulas for overlap integrals over four-component relativistic Slater spinor orbitals with the same screening constants in position space are also derived. The relations obtained in this study can be useful in the study of different problems arising in the quasirelativistic and relativistic quantum mechanics when the position, momentum and four dimensional spaces are employed.

Abstract:
We calculate the single-valued and spinor representations of $SO_4$, the orientation-preserved subgroup of $O_4$, on the base of our previous work on four-dimensional cubic group $O_4$.

Abstract:
Spinor and twistor formulations of tensionless bosonic strings in 4-dimensional Minkowski space are constructed. We begin with a first-order action that is equivalent to the Nambu-Goto action in the tensionful case and that leads to a spinorial action in the tensionless case. From this spinorial action, we find an alternative spinorial action useful for constructing a simple twistor formulation of tensionless strings. The twistor formulation is steadily constructed in accordance with a fundamental concept of twistor theory. We investigate local internal symmetries inherent in the twistorial action for a tensionless string and carry out some classical analyses of the tensionless string expressed in a twistorial form.

Abstract:
The analytical relations in position, momentum and four-dimensional spaces are established for the expansion and one-range addition theorems of relativistic complete orthonormal sets of exponential type spinor wave functions and Slater spinor orbitals of arbitrary half-integral spin. These theorems are expressed through the corresponding nonrelativistic expansion and one-range addition theorems of the spin-0 particles introduced by the author. The expansion and one-range addition theorems derived are especially useful for the computation of multicenter integrals over exponential type spinor orbitals arising in the generalized relativistic Dirac-Hartree-Fock-Roothaan theory when the position, momentum and four-dimensional spaces are employed.

Abstract:
Using the complete orthonormal sets of radial parts of nonrelativitistic exponential type orbitals (2,1, 0, 1, 2, ...) and spinor type tensor spherical harmonics of rank s the new formulae for the 2(2s+1)-component relativistic spinors useful in the quantum mechanical description of the arbitrary half-integral spin particles by the generalized Dirac equation introduced by the author are established in position, momentum and four-dimensional spaces, where 1/ 2, 3 / 2, 5 / 2, ... s = . These spinors are complete without the inclusion of the continuum. The 2(2s+1)component spinors obtained are reduced to the independent sets of two-component spinors defined as a product of complete orthonormal sets of radial parts of orbitals and twocomponent spinor type tensor spherical harmonics. We notice that the new idea presented in this work is the unified treatment of half-integral spin and scalar particles in position, momentum and four-dimensional spaces. Relations presented in this study can be useful in the linear combination of atomic orbitals approximation for the solution of different problems arising in the relativistic quantum mechanics when the orthonormal basis sets of relativistic exponential type spinor wave functions and Slater type spinor orbitals in position, momentum and four -dimensional spaces are employed.

Abstract:
Microbial community patterns vary in glaciers world wide, presenting unique responses to global climatic and environmental changes. Four bacterial clone libraries were established by 16S rRNA gene amplification from four ice layers along the 42-m-long ice core MuztB drilled from the Muztag Ata Glacier. A total of 152 bacterial sequences obtained from the ice core MuztB were phylogenetically compared with the 71 previously reported sequences from three ice cores extracted from ice caps Malan, Dunde, and Puruoganri. The six functional clusters Flavisolibacter, Flexibacter (Bacteroidetes), Acinetobacter, Enterobacter (Gammaproteobacteria), Planococcus/Anoxybacillus (Firmicutes), and Propionibacter/Luteococcus (Actinobacteria) frequently occurred along the Muztag Ata Glacier profile. Sequence analysis showed that most of the sequences from the ice core clustered with those from cold environments, and the sequences from the same glacier formed a distinct cluster. Moreover, bacterial communities from the same location or similarly aged ice formed a cluster, and were clearly separate from those from other geographically isolated glaciers. In a summary, the findings provide preliminary evidence of zone distribution of microbial community, support our hypothesis of the spatial and temporal biogeography of microorganisms in glacial ice.

Abstract:
Anti-self-dual metrics in the $(++--)$ signature which admit a covariantly constant real spinor are studied. It is shown that finding such metrics reduces to solving a fourth order integrable PDE, and some examples are given. The corresponding twistor space is characterised by existence of a preferred non-zero real section of $\kappa^{-1/4}$, where $\kappa$ is the canonical line bundle of the twistor space It is demonstrated that if the parallel spinor is preserved by a Killing vector, then the fourth order PDE reduces to the dispersionless Kadomtsev--Petviashvili equation and its linearisation. Einstein--Weyl structures on the space of trajectories of the symmetry are characterised by the existence of a parallel weighted null vector.

Abstract:
I introduce a spinor field theory for the photon. The three-dimensional vector electromagnetic field and the four-dimensional vector potential are components of this spinor photon field. A spinor equation for the photon field is derived from Maxwell's equations,the relations between the electromagnetic field and the four-dimensional vector potential, and the Lorentz gauge condition. The covariant quantization of free photon field is done, and only transverse photons are obtained. The vacuum energy divergence does not occur in this theory. A covariant "positive frequency" condition is introduced for separating the photon field from its complex conjugate in the presence of the electric current and charge.

Abstract:
A representation of the Lorentz group is given in terms of 4 X 4 matrices defined over the hyperbolic number system. The transformation properties of the corresponding four component spinor are studied, and shown to be equivalent to the transformation properties of the complex Dirac spinor. As an application, we show that there exists an algebra of automorphisms of the complex Dirac spinor that leaves the transformation properties of its eight real components invariant under any given Lorentz transformation. Interestingly, the representation of the Lorentz algebra presented here is naturally embedded in the Lie algebra of a group isomorphic to SO(3,3;R) instead of the conformal group SO(2,4;R).