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It is well known that ‘an almost complex structure’ J that is J² = −I on the manifold M is called ‘an almost Hermitian manifold’ (M, J, G) if G(JX, JY) = G(X, Y) and proved that (F²M, JD, GD) is ‘an almost Hermitian manifold’ on the frame bundle of the second order F²M. The term ‘an almost complex structure’ refers to the general quadratic structure J² = pJ + qI, where p = 0, q = −1. However, this paper aims to study the general quadratic equation J² = pJ + qI, where p, q are positive integers, it is named as a metallic structure. The diagonal lift of the metallic structure J on the frame bundle of the second order F²M is studied and shows that it is also a metallic structure. The proposed theorem proves that the diagonal lift GD of a Riemannian metric G is a metallic Riemannian metric on F²M. Also, a new tensor field ˜J of type (1,1) is defined on F²M and proves that it is a metallic structure. The 2-form and its derivative dF of a tensor field ˜J are determined. Furthermore, the Nijenhuis tensor N˜J of a metallic structure ˜J and the Nijenhuis tensor N J D of a tensor field JD of type (1,1) on the frame bundle of the second order F²M are calculated.

Purpose This paper aims to examine three important questions: What would be the effects of pricing at the lower end of a wide vs narrow latitude of price acceptance (LPA) on consumer choice of the bundle? How would the nature of a bundle frame (i.e. discount on bundle vs discount on components) and discount frame (i.e. discount as absolute off vs discount as percentage off) influence the preference given to a price level that is at the wide or narrow end of the LPA? Would the effect be significantly different if the bundle components were complementary vs if they were non-complementary? Design/methodology/approach The authors carried out two studies using between-subject experimental design. In Study 1, the authors used 2 (LPA: wide/narrow) × 2 (complementarity: yes/no) × 2 (bundle frame: together/separate) design, and in Study 2, the authors replaced bundle frame with discount frame (i.e. absolute off/percentage off). Findings The authors find that the LPA effect is likely to outweigh the complementarity effect; however, a combined effect of complementarity and bundle frame is stronger than the LPA effect. Also, for a wide (narrow) LPA product bundle, absolute off (percentage off) discount frame is more attractive. Practical implications Managers should use bundling strategy with complementary products having wider LPA. In case of wide LPA and complementary products, both together and separate frame could be the best bundling strategy while in case of narrow LPA and complementary products, together frame could be the best bundling strategy. Originality/value The main contribution relates to the role LPA plays in consumer evaluation of a bundle offer and its interaction with complementarity and discount frame. The authors apply the range hypothesis principles (i.e. price-attractiveness judgments are based on a comparison of market prices to the endpoints of a range of evoked prices) in the bundling context and extend the earlier work in the area of complementarity and discount frame.

In this work, we have characterized the frame bundle FM admitting metallic structures on almost quadratic ϕ-manifolds ϕ2=pϕ+qI−qη⊗ζ, where p is an arbitrary constant and q is a nonzero constant. The complete lifts of an almost quadratic ϕ-structure to the metallic structure on FM are constructed. We also prove the existence of a metallic structure on FM with the aid of the J˜ tensor field, which we define. Results for the 2-Form and its derivative are then obtained. Additionally, we derive the expressions of the Nijenhuis tensor of a tensor field J˜ on FM. Finally, we construct an example of it to finish.

Maximal couplings are (probabilistic) couplings of Markov processes such that the tail probabilities of the coupling time attain the total variation lower bound (Aldous bound) uniformly for all time. Markovian (or immersion) couplings are couplings defined by strategies where neither process is allowed to look into the future of the other before making the next transition. Markovian couplings are typically easier to construct and analyze than general couplings, and play an important role in many branches of probability and analysis. Hsu and Sturm, in a preprint circulating in 2007, but later published in 2013, proved that the reflection-coupling of Brownian motion is the unique Markovian maximal coupling ( MMC ) of Brownian motions starting from two different points. Later, Kuwada (Electron J Probab 14(25), 633–662, 2009 ) proved that the existence of a MMC for Brownian motions on a Riemannian manifold enforces existence of a reflection structure on the manifold. In this work, we investigate suitably regular elliptic diffusions on manifolds, and show how consideration of the diffusion geometry (including dimension of the isometry group and flows of isometries) is fundamental in classification of the space and the generator of the diffusion for which an MMC exists, especially when the MMC also holds under local perturbations of the starting points for the coupled diffusions. We also describe such diffusions in terms of Killing vectorfields (generators of isometry groups) and dilation vectorfields (generators of scaling symmetry groups). This permits a complete characterization of those possible manifolds and their diffusions for which there exists a MMC under local perturbations of the starting points of the coupled diffusions. For example, in the time-homogeneous case it is shown that the only possible manifolds that may arise are Euclidean space, hyperbolic space and the hypersphere. Moreover the permissible drifts can then derive only from rotation isometries of these spaces (and dilations, in the Euclidean case). In this sense, a geometric rigidity phenomenon holds good.

We provide a probabilistic and infinitesimal view of how the principal component analysis procedure (PCA) can be generalized to analysis of nonlinear manifold valued data. Starting with the probabilistic PCA interpretation of the Euclidean PCA procedure, we show how PCA can be generalized to manifolds in an intrinsic way that does not resort to linearization of the data space. The underlying probability model is constructed by mapping a Euclidean stochastic process to the manifold using stochastic development of Euclidean semimartingales. The construction uses a connection and bundles of covariant tensors to allow global transport of principal eigenvectors, and the model is thereby an example of how principal fiber bundles can be used to handle the lack of global coordinate system and orientations that characterizes manifold valued statistics. We show how curvature implies nonintegrability of the equivalent of Euclidean principal subspaces, and how the stochastic flows provide an alternative to explicit construction of such subspaces. We describe estimation procedures for inference of parameters and prediction of principal components, and we give examples of properties of the model on embedded surfaces.

We consider two principal bundles of embeddings with total space E m b ( M , N ) , with structure groups D i f f ( M ) and D i f f + ( M ) , where D i f f + ( M ) is the groups of orientation preserving diffeomorphisms. The aim of this paper is to describe the structure group of the tangent bundle of the two base manifolds: B ( M , N ) = E m b ( M , N ) / D i f f ( M ) and B + ( M , N ) = E m b ( M , N ) / D i f f + ( M ) from the various properties described, an adequate group seems to be a group of Fourier integral operators, which is carefully studied. It is the main goal of this paper to analyze this group, which is a central extension of a group of diffeomorphisms by a group of pseudo-differential operators which is slightly different from the one developped in the mathematical litterature e.g. by H. Omori and by T. Ratiu. We show that these groups are regular, and develop the necessary properties for applications to the geometry of B ( M , N ) . A case of particular interest is M = S 1 , where connected components of B + ( S 1 , N ) are deeply linked with homotopy classes of oriented knots. In this example, the structure group of the tangent space T B + ( S 1 , N ) is a subgroup of some group G L r e s , following the classical notations of (Pressley, A., 1988). These constructions suggest some approaches in the spirit of one of our previous works on Chern-Weil theory that could lead to knot invariants through a theory of Chern-Weil forms.

We describe how the kinematic system of rolling two $n$-dimensional connected, oriented Riemannian manifolds $M$ and $\\widehat M$ without twisting or slipping can be lifted to a nonholonomic system defined on the product of the oriented orthonormal frame bundles belonging to the two manifolds. By using known properties of forms known as Cartan's moving frame, we obtain sufficient conditions for the local controllability of the system in terms of the curvature tensors and the sectional curvatures of the manifolds involved. By using the information from these calculations, we show that we need only consider normal extremals, when looking for a rolling of minimal length, connecting two given configurations. We also give some results for controllability in the particular cases when $M$ and $\\widehat M$ are locally symmetric or complete. [PUBLICATION ABSTRACT]

The paper is focused on mathematical approach to materials with microstructure, the special case of them are Cosserat media. Frame bundles are described with respect to their role in continuum mechanics and the structure jet groups. General (r-th order) holonomic, nonholonomic and semiholonomic microstructure configuration are introduces and the algebraic approach using Weil algebras is presented, too.

A manifold with an arbitrary affine connection is considered and the geodesic spray associated with the connection is studied in the presence of a Lie group action. In particular, results are obtained that provide insight into the structure of the reduced dynamics associated with the given invariant affine connection. The geometry of the frame bundle of the given manifold is used to provide an intrinsic description of the geodesic spray. A fundamental relationship between the geodesic spray, the tangent lift and the vertical lift of the symmetric product is obtained, which provides a key to understanding reduction in this formulation.

We investigate the curvature of the so-called diagonal lift from an affine manifold to the linear frame bundle LM. This is an affine analogue (but not a direct generalization) of the Sasaki-Mok metric on LM investigated by L.A. Cordero and M. de León in 1986. The Sasaki-Mok metric is constructed over a Riemannian manifold as base manifold. We receive analogous and, surprisingly, even stronger results in our affine setting.