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For $M$ a closed manifold or the Euclidean space $\\mathbb{R}^n$, the authors present a detailed proof of regularity properties of the composition of $H^s$-regular diffeomorphisms of $M$ for $s >\\frac{1}{2}\\dim M 1$.

In this paper, we study the centralizer of a partially hyperbolic diffeomorphism on${\\mathbb T}^3$which is homotopic to an Anosov automorphism, and we show that either its centralizer is virtually trivial or such diffeomorphism is smoothly conjugate to its linear part.

We prove a homological stability theorem for moduli spaces of manifolds of dimension 2n, for attaching handles of index at least n, after these manifolds have been stabilised by countably many copies of Sⁿ × Sⁿ. Combined with previous work of the authors, we obtain an analogue of the Madsen–Weiss theorem for any simply-connected manifold of dimension 2n ≥ 6.

We give a $C^1$ -perturbation technique for ejecting an a priori given finite set of periodic points preserving a given finite set of homo/heteroclinic intersections from a chain recurrence class of a periodic point. The technique is first stated under a simpler setting called a Markov iterated function system, a two-dimensional iterated function system in which the compositions are chosen in a Markovian way. Then we apply the result to the setting of three-dimensional partially hyperbolic diffeomorphisms.

Producing spatial transformations that are diffeomorphic is a key goal in deformable image registration. As a diffeomorphic transformation should have positive Jacobian determinant |J| everywhere, the number of pixels (2D) or voxels (3D) with |J|<0 has been used to test for diffeomorphism and also to measure the irregularity of the transformation. For digital transformations, |J| is commonly approximated using a central difference, but this strategy can yield positive |J|’s for transformations that are clearly not diffeomorphic—even at the pixel or voxel resolution level. To show this, we first investigate the geometric meaning of different finite difference approximations of |J|. We show that to determine if a deformation is diffeomorphic for digital images, the use of any individual finite difference approximation of |J| is insufficient. We further demonstrate that for a 2D transformation, four unique finite difference approximations of |J|’s must be positive to ensure that the entire domain is invertible and free of folding at the pixel level. For a 3D transformation, ten unique finite differences approximations of |J|’s are required to be positive. Our proposed digital diffeomorphism criteria solves several errors inherent in the central difference approximation of |J| and accurately detects non-diffeomorphic digital transformations. The source code of this work is available at https://github.com/yihao6/digital_diffeomorphism.

For every $k \\geq 2$ and $n \\geq 2$ , we construct n pairwise homotopically inequivalent simply connected, closed $4k$ -dimensional manifolds, all of which are stably diffeomorphic to one another. Each of these manifolds has hyperbolic intersection form and is stably parallelisable. In dimension four, we exhibit an analogous phenomenon for spin $^{c}$ structures on $S^2 \\times S^2$ . For $m\\geq 1$ , we also provide similar $(4m-1)$ -connected $8m$ -dimensional examples, where the number of homotopy types in a stable diffeomorphism class is related to the order of the image of the stable J-homomorphism $\\pi _{4m-1}(SO) \\to \\pi ^s_{4m-1}$ .

Anosov automorphisms with Jordan blocks are not periodic data rigid. We introduce a refinement of the periodic data and show that this refined periodic data characterizes $C^{1+}$ conjugacy for Anosov automorphisms on $\\mathbb {T}^4$ with a Jordan block.

Our paper presents a nonparametric data-driven technique that can enhance the accuracy of robot kinematics models by reducing geometric and nongeometric inaccuracies. We propose this approach based on the theory of singular maps and the Large Dense Diffeomorphic Metric Mapping (LDDMM) framework, which has been developed in the field of Computational Anatomy. This framework can be thought of as a method for identifying nonlinear static models that encode a priori knowledge as a nominal model that we deform using diffeomorphisms. To tackle the kinematic calibration problem, we implement Calibration by Diffeomorphisms and obtain a solution using an image registration formalism. We evaluate our approach via simulations on double pendulum robot models, which account for both geometric and nongeometric discrepancies. The simulations demonstrate an improvement in the precision of the kinematics results for both types of inaccuracies. Additionally, we discuss the potential application of physical experiments. Our approach provides a fresh perspective on robot kinematics calibration using Calibration by Diffeomorphisms, and it has the potential to address inaccuracies caused by unknown or difficult-to-model phenomena.

We consider the minimization of a functional of the calculus of variations, under assumptions that are diffeomorphism invariant. In particular, a nonuniform coercivity condition needs to be considered. We show that the direct methods of the calculus of variations can be applied in a generalized Sobolev space, which is in turn diffeomorphism invariant. Under a suitable (invariant) assumption, the minima in this larger space belong to a usual Sobolev space and are bounded.