Covariant Metrics Research Articles

Gas Turbine Bearing Temperature Monitoring via Regression Modelling

This paper focuses on using Regression technique (MLR) towards finding solution to incidence of high compressor bearing temperature on one of the units at Geregu power plant in Ajaokuta, Nigeria. Monitoring of parameters related to the bearing temperature was carried out to find out causes for the high bearing temperature fault and came up with successful diagnosis by interrelating the gas turbine current lube oil test results of parameters like the kinematic viscosities, % concentration of additives and flash point with reference and standard VG46 lube oil data published in literature. Using statistical tools like the Pearson correlation and co-variant metrics for the five-years, the oil viscosities at 100oC and 40oC were selected as the input of the MLR model based on their Pearson coefficients of (-98.08%) and (-99.68%) respectively relative to the compressor bearing temperature. The MLR model for the bearing temperature prediction gave a root mean square error of 0.121 and coefficient of determination (R2) of 99.71%. The model predicts that by the 2nd quarter of 2025, the bearing temperature would have reached the alarm point (900C) from the current value of 850C and that by the 1st quarter of 2027, the bearing temperature would have reached the trip point (1200C). Conclusion reached is that a well formulated data driven model can reliably forecast bearing temperature and together with sensors aid in gas turbine condition monitoring. Likewise, it is concluded that shearing due to the consistent high temperature operation of the gas turbine lube oil is responsible for the depletion of the Zinc (-23.9%) and Magnesium(-26%) additives leading to the decay in the viscosity and consequent bearing temperature increment. Recommendation made is to either replenish oil with anti-wear additives or completely replace the oil to minimize the bearing wear rate and thus the bearing temperature.

Braided Cartan calculi and submanifold algebras

We construct a noncommutative Cartan calculus on any braided commutative algebra and study its applications in noncommutative geometry. The braided Lie derivative, insertion and de Rham differential are introduced and related via graded braided commutators, also incorporating the braided Schouten–Nijenhuis bracket. The resulting braided Cartan calculus generalizes the Cartan calculus on smooth manifolds and the twisted Cartan calculus. While it is a necessity of derivation based Cartan calculi on noncommutative algebras to employ central bimodules our approach allows to consider bimodules over the full underlying algebra. Furthermore, equivariant covariant derivatives and metrics on braided commutative algebras are discussed. In particular, we prove the existence and uniqueness of an equivariant Levi-Civita covariant derivative for any fixed non-degenerate equivariant metric. Operating in a symmetric braided monoidal category we argue that Drinfel’d twist deformation corresponds to gauge equivalences of braided Cartan calculi. The notions of equivariant covariant derivative and metric are compatible with the Drinfel’d functor as well. Moreover, we project braided Cartan calculi to submanifold algebras and prove that this process commutes with twist deformation.

Linear Multiscale Analysis of Similarities between Images on Riemannian Manifolds: Practical Formula and Affine Covariant Metrics

In this paper we study the problem of comparing two patches of images defined on Riemannian manifolds which in turn can be defined by each image domain with a suitable metric depending on the image. For that we single out one particular instance of a set of models defining image similarities that was earlier studied in [C. Ballester et al., Multiscale Model. Simul., 12 (2014), pp. 616--649], using an axiomatic approach that extended the classical Álvarez--Guichard--Lions--Morel work to the nonlocal case. Namely, we study a linear model to compare patches defined on two images in $\mathbb{R}^N$ endowed with some metric. Besides its genericity, this linear model is selected by its computational feasibility since it can be approximated leading to an algorithm that has the complexity of the usual patch comparison using a weighted Euclidean distance. Moreover, we propose and study some intrinsic metrics which we define in terms of affine covariant structure tensors and we discuss their properties. These tensors are defined for any point in the image and are intrinsically endowed with affine covariant neighborhoods. We also discuss the effect of discretization over the affine covariance properties of the tensors. We illustrate our theoretical results with numerical experiments.

Formula omitted]-compatible connections in noncommutative Riemannian geometry

We develop the formalism for noncommutative differential geometry and Riemmannian geometry to take full account of the ∗ -algebra structure on the (possibly noncommutative) coordinate ring and the bimodule structure on the differential forms. We show that ∗ -compatible bimodule connections lead to braid operators σ in some generality (going beyond the quantum group case) and we develop their role in the exterior algebra. We study metrics in the form of Hermitian structures on Hilbert ∗ -modules and metric compatibility in both the usual form and a cotorsion form. We show that the theory works well for the quantum group C q [ S U 2 ] with its three-dimensional calculus, finding for each point of a three-parameter space of covariant metrics a unique ‘Levi-Civita’ connection deforming the classical one and characterised by zero torsion, metric preservation and ∗ -compatibility. Allowing torsion, we find a unique connection with a classical limit that is metric preserving and ∗ -compatible and for which σ obeys the braid relations. It projects to a unique ‘Levi-Civita’ connection on the quantum sphere. The theory also works for finite groups, and in particular for the permutation group S 3 , where we find somewhat similar results.

ELLIPTIC CURVES, ALGEBRAIC GEOMETRY APPROACH IN GRAVITY THEORY AND UNIFORMIZATION OF MULTIVARIABLE CUBIC ALGEBRAIC EQUATIONS

Based on the distinction between the covariant and contravariant metric tensor components in the framework of the affine geometry approach and the so called "gravitational theories with covariant and contravariant connection and metrics", it is shown that a wide variety of 3rd, 4th, 5th, 7th, 10th- degree algebraic equations exists in gravity theory. This is important in view of finding new solutions of the Einstein's equations, if they are treated as algebraic ones. Since the obtained cubic algebraic equations are multivariable, the standard algebraic geometry approach for parametrization of two-dimensional cubic equations with the elliptic Weierstrass function cannot be applied. Nevertheless, for a previously considered cubic equation for reparametrization invariance of the gravitational Lagrangian and on the base of a newly introduced notion of "embedded sequence of cubic algebraic equations", it is demonstrated that in the multivariable case such a parametrization is also possible, but with complicated irrational and non-elliptic functions. After finding the solutions of a system of first-order nonlinear differential equations, these parametrization functions can be considered also as uniformization ones (depending only on the complex uniformization variable z) for the initial multivariable cubic equation.

Spaces with contravariant and covariant affine connections and metrics

The theory of spaces with contravariant and covariant affine connections, whose components differ not only in sign, and metrics [(L n ,g) spaces] is worked out within the framework of tensoranalysis over differentiable manifolds and in a volume necessary for further consideration of the kinematics of vector fields and the Lagrangiantheory of tensor fields over (L n ,g) spaces. The possibility of introducing affine connections for contravariant and covariant tensor fields, whose components differ not only in sign, over differentiable manifolds with finite dimensions is discussed. The action of the deviation operator, having an important role for deviation equations in gravitational physics, is considered for the case of contravariant and covariant vector fields over differentiable manifolds with different affine connections (called L n spaces). A deviation identity for contravariant vector fields is obtained. The notions of covariant, contravariant, covariant projective, and contravariant projective metric are introduced in (L n ,g) spaces. The action of the covariant and the Lie differential operator on the different types of metric is found. The concepts of symmetric covariant and contravariant (Riemannian) connection are defined and presented by means of the covariant and contravariant metric and the corresponding torsion tensors. The different types of relative tensor fields (tensor densities) as well as the invariant differential operators acting on them are considered. The invariant volume element and its properties under the action of different differential operators are investigated.

Lagrangian Theory of Tensor Fields over Spaces with Contravariant and Covariant Affine Connections and Metrics and Its Application to Einstein's Theory of Gravitation in V 4 Spaces

The Lagrangian formalism for tensor fields over differentiable manifolds with contravariant and covariant affine connections (whose components differ not only by sign) and a metric \([(\bar L_n ,g) - spaces]\) is considered. The functional, the Lie, the covariant, and the total variations of a Lagrangian density, depending on components of tensor fields (with finite rank) and their first and second covariant derivatives, are established. A variation operator is determined and the corollaries of its commutation relations with the covariant and the Lie differential operators are found. The canonical (common) method of Lagrangians with partial derivatives (MLPD) and the method of Lagrangians with covariant derivatives (MLCD) are outlined. They differ each other by the commutation relations the variation operator has to obey with the covariant and the Lie differential operator. The covariant Euler–Lagrange equations are found on the basis of the MLCD. The energy-momentum tensors are considered on the basis of the Lie variation and the covariant Noether identities.

FERMI DERIVATIVE AND FERMI–WALKER TRANSPORTS OVER $(\bar{L}_n, g)$-SPACES

The notions of Fermi covariant differential operator, Fermi derivative, and Fermi–Walker transport are generalized for the case of differentiable manifolds with different (not only by sign) contravariant and covariant affine connections and metrics [[Formula: see text]-spaces]. The generalization of Fermi–Walker transport is not unique and depends on the structure of the covariant antisymmetric tensor of second rank in the construction of the Fermi–Walker transport. The existence of such type of transport over [Formula: see text]-spaces allows the determination of a proper nonrotating accelerated observer's frame of reference, if a [Formula: see text]-space is used as a model of the space–time.

Generally covariant constitutive equations and optical metrics

The generally covariant constitutive equations and the corresponding optical metrics for a moving anisotropic medium are first derived. The obtained results reduce to the previous ones for a moving isotropic medium. An extension to any curved space-time is examined.

Post-Newtonian expansion of an algebraically extended theory of gravity

The post-Newtonian expansion is analyzed for an algebraically extended theory of gravity equivalent to a theory with a real, nonsymmetric metric, previously referred to as Algebraically Extended Hilbert Gravity (AHG). The hermiticity of the algebra-valued covariant and contravariant metrics is found to constrain the form of the expansion of the antisymmetric part of the real metric to one of two possibilities. Both of these display a qualitatively new technical feature: the lowest order equations are nonlinear, depriving the post-Newtonian expansion of its greatest asset. In one case, they lead to a solution with a negative Newtonian mass parameter. In the other case, they lead to a contact-type solution in which some metric components depend directly upon the stress–energy tensor rather than the integral of this distribution. The unphysical nature of these solutions rules out AHG as a physically viable alternative theory of gravitation.

Symmetry and internal time on the superspace of asymptotically flat geometries.

A difficulty with the canonical approach to quantum gravity, leading to attempts at "third quantization," is the absence of symmetry vectors on the superspace of three-metrics: vector fields that generate transformations of superspace leaving the action invariant. We show that on the superspace of asymptotically flat three-metrics, such symmetry vectors exist. They correspond to diffeomorphisms of each three-geometry that behave asymptotically as elements of the symmetry group at spatial infinity. The conserved momentum associated with a symmetry vector has a conjugate variable which can be regarded as an internal time coordinate of an isolated system. In particular, for asymptotic translations, a corresponding internal time is a center-of-mass coordinate. An appendix considers the natural contravariant and covariant metrics on superspace. Because natural contravariant metrics are not everywhere invertible, the associated covariant metrics are not everywhere defined.