Reconstruction theorem for groupoids and principal fiber bundles

The loop space formulation of 3+1 canonical quantum gravity premises that all physical information is contained within the holonomy loop functionals. This assumption is the result of the reconstruction theorem for a principla fiber bundle on a base loop space. The gauge connection for interacting gauge theories is more appropriately and readily reconstructed on a path space as opposed to a loop space. We generalize the reconstruction theorem to a base path space. Employing a holonomy groupoid map and a path connection, we trivially construct an abstract Lie groupoid from which a principal fiber bundle and gauge connection can be derived as distinctive examples. The groupoid reconstruction theorem is valid on both connected and nonconnected base manifolds, unlike the holonomy group reconstruction theorem, which can only be utilized for connected manifolds.     

Variational principles in the theory of gravitoinertial fields

Variational methods and some features of their application in the theory of gravitoinertial fields are considered.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 110–114, November, 1974.     

Quantization of fields on principal bundles

A constructive method of quantization of free gauge fields is presented. In particular, by using a geometric approach (classical gauge fields as connections on principal fibre bundles) and the Borchers-Uhlmann method, an algebraic quantum field theory of gauge fields is constructed. Thus, a linearized version of a quantum field theory of non-Abelian gauge fields is obtained generalizing Bongaarts' results for the Abelian U(1)-theory.Invited talk presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.     

Group actions on principal bundles and dimensional reduction

Using the invariant geometrical interpretation of gauge and Higgs fields, a simple derivation is given of the dimensional reduction procedure. The underlying assumption with regard to the Riemannian structure, group orbits and invariant connection are clarified and the critical points of the Higgs potential are shown to have a natural geometrical interpretation.     

A note on first order differential calculus on quantum principal bundles

The relationship between the exactness of a first order differential calculus on a comodule algebra P and the Galois property of P is investigated.     

Variational principles for multi-walled carbon nanotubes undergoing buckling based on nonlocal elasticity theory

Variational principles are derived for multi-walled carbon nanotubes undergoing buckling using the semi-inverse method. Derivations are based on the continuum modelling of nanotubes taking into account small scale effects via the nonlocal theory of elasticity. Natural and geometric boundary conditions for multi-walled nanotubes are derived which leads to a set of coupled boundary conditions.     

Almost complex structures on cotangent bundles and generalized geometry

We study a class of complex structures on the generalized tangent bundle of a smooth manifold M$M$ endowed with a torsion free linear connection, ∇$\nabla$. We introduce the concept of ∇$\nabla$-integrability and we study integrability conditions. In the case of the generalized complex structures introduced by Hitchin (2003) in [2], we compare the two concepts of integrability. Moreover, as an application, we describe almost complex structures on the cotangent bundle of M$M$ induced by complex structures on the generalized tangent bundle of M$M$.     

Variational principles of electrostatics and higher polarizabilities of conductors

Based on variational principles, new concepts of higher polarizability of an isolated conductor are introduced that generalize well-known concepts of capacity and polarizability. It is shown that the lth-order polarizability in a spherical coordinate system is represented by a (2l+1) × (2l+1) matrix. The corresponding (2l+1) potentials can be considered as a system of lth-order multipoles typical of a conductor and more general than systems of spherical multipoles known in electrodynamics. It is exemplified what new results can be obtained using the higher polarizability technique for solving problems in electrostatics. Krasnoyarsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 61–65, October, 1999.     

Grading of spinor bundles and gravitating matter in noncommutative geometry

The gravitating matter is studied within the framework of noncommutative geometry. The noncommutative Einstein-Hilbert action on the product of a four-dimensional manifold with discrete space gives models of matter fields coupled to the standard Einstein gravity. The matter multiplet is encoded in the Dirac operator which yields a representation of the algebra of universal forms. The general form of the Dirac operator depends on a choice of the grading of the corresponding spinor bundle. A choice is given, which leads to the nonlinear vectorσ-model coupled to the Einstein gravity.     

Eigenforms of the spin Laplacian and projectable spinors for principal bundles

Let π : P → Y be a principal bundle with compact connected structure group G over a compact spin manifold Y. We use a suitably chosen invariant spinor on G to define pull-back operator from the spin bundle on Y to the spin bundle on P and study when the pull-back of an eigenspinor on Y is an eigenspinor on P.     

Variational principles and the effect of a cutoff on population pattern size

The relation between pattern size and maximum population density is obtained for the stationary state of populations living in a refuge surrounded by a hostile environment. The population dynamics is described by reaction–diffusion equations whose kinetic terms display a cutoff. The latter takes into account the discreteness of the population when the population density is small. We employ a variational principle for the nonlinear eigenvalue problem to obtain lower bounds for the pattern length. Numerical solutions display excellent agreement with our analytical results.     

Basic principles of the surface harmonics method: Flat geometry

The basic principles of the surface harmonics method are described. A one-dimensional problem is used to exemplify the specific features of the method and the algorithms for construction of finite-difference equations. The objective of this study is to popularize the surface harmonics method among specialists.     

Physical geometry: A unified theory of gravitation,electromagnetism and other interactions

An invariant correlation and a variational principle are given for the theory of connections and frames introduced in previous papers. The relation of the resultant gravitation theory to Yang's theory is clarified. The resultant equations of motion, which imply a generalized Dirac equation, are used to understand geometrically certain aspects of relativistic quantum theory. The conjecture is proposed that electrornegnetism is related to anSU(2) subgroup. The possible association of the extra generators with strong and weak nuclear forces is discussed.     

Lattice gauge fields,principal bundles and the calculation of topological charge

SU₂-valued lattice gauge fields are studied on a 4-dimensional simplicial lattice. Ifu has sufficiently small plaquette products, then there is a unique principal SU₂-bundle admitting transition functions, defined on the intersections of adjacent dual cells, which take values within /8 ofu. An algorithm is explicitly given which associates an integer to everyu off a certain set of measure zero. This algorithm only involves evaluation of 4×4 determinants and the solution of quadratic equations. Whenu is as above, the integer produced is the second Chern number of , i.e. the topological charge ofu.     

Hollow Bragg fiber bundles: when coupling helps and when it hurts

Coupling between leaky modes of collinear hollow-core Bragg fibers is considered. It is found that coupling is unusually strong because of resonant effects in the interfiber cavity when the fibers are touching each other and decreases dramatically with the first tens of nanometers of fiber separation. However, residual coupling with a strength proportional to the fiber radiation loss is long range, decreasing as an inverse square root of the interfiber separation and exhibiting periodic variation. The possibility of building a directional coupler from touching Bragg fibers is discussed in view of the findings.     

Galloping of iced quad-conductors bundles based on curved beam theory

Galloping refers to wind-induced, low-frequency, large-amplitude oscillations that have been more frequently observed for a bundle conductor than for a single conductor. In the present work two different models are built to investigate the galloping of a bundle conductor: (1) a finite curved beam element method and (2) a hybrid model based on curved beam element theory. The finite curved beam element model is effective in dealing with the spacers between the bundled conductors and the joint between the conductors and spacers that can be simulated as a rigid joint or a hinge. Furthermore, the finite curved beam element model can be used to deal with large deformation. The hybrid model invokes the small deformation hypothesis and has a high computational efficiency. A hybrid model based on conventional cable element theory is also programmed to be compared with the aforementioned models based on curved beam element theory. Numerical examples are presented to assess the accuracy of the different models in predicting the equilibrium conductor position, natural frequencies and galloping amplitude. The results show that the curved beam element models, involving more degrees of freedom and coupling of translational and torsional motion, are more accurate at simulating the static and dynamic characters of an iced quad-conductor bundle. The use of hinges, rather than rigid connections, reduces the structural response amplitudes of a galloping conductor bundle.     

Inertia and damping effects on the damage-dependent burst size distribution of fiber bundles

Distributions of simultaneous fiber failures—bursts—in loaded fiber bundles are studied considering inertia and damping. Resulting burst size distributions have universal properties: all approach the power law D_ΔΔ^−2.5 for larger burst sizes Δ. Momentary burst size distributions evolve with increasing damage and do not follow power laws but are still universal. Finally, it is briefly outlined how to use distribution progression to assess damage state.