Search Results (18)

Lightweight flapping-wing robots are affected by structural flexibility, wind disturbances, and static friction in basal passive joints during perching and attitude-holding tasks. These coupled effects can make conventional PID and sliding mode control (SMC) produce error amplification, torque fluctuation, flexible-response excitation, and attitude-boundary violation. This study establishes an ADAMS–Simulink co-simulation platform for a rigid–flexible coupled flapping-wing robot and proposes a diffeomorphism-based attitude-constrained controller. The inverse hyperbolic tangent mapping transforms bounded physical errors into unbounded virtual errors, allowing smooth small-error regulation and stronger constraint enforcement near safety boundaries. Wind-free tracking, compound wind rejection, pulse wind scanning, mapping-parameter sensitivity, and a CBF-QP safety-filtered baseline are evaluated. In manuscript parameter-synchronized ADAMS 2024 reruns under a 2 m/s steady wind with a 1 m/s pulse, PID and SMC show runaway angular excursions of 1602.56° and 381,330.03°, whereas the proposed method remains bounded at 23.58° with an RMSE of 2.943° and no boundary violation. The CBF-QP baseline still violates the boundary at 1394°. The results show improved tracking accuracy, boundary protection, measured-channel flexible-excitation attenuation, and stable disturbance recovery. Full article

We give a proof of Cartan’s Theorem on totally geodesic submanifolds for real analytic manifolds endowed with a real analytic, torsion-free, affine connection. We apply the theorem to real analytic Hadamard manifolds and, more generally, to real analytic manifolds with a torsion-free, analytic, affine connection, such that at a manifold point $p\in M$, the exponential map is a real analytic diffeomorphism from the tangent space $T_p\left(M\right)$ to M. Examples of manifolds with this property are statistical manifolds with a cubic form divisible by the metric, as was recently proven. We also give examples of totally geodesic submanifolds obtained as fixed points of affine transformations of M and, moreover, as certain submanifolds of connected Lie groups with the 0-connection of Cartan–Schouten. Finally, we also determine all connected complete totally geodesic surfaces of the Riemannian manifold $({\mathcal{P}}_2,g)$ of symmetric positive definite $2\times 2$ real matrices, endowed with the trace metric g. Full article

This work formulates semiclassical gravity within a causal-diamond framework where a finite-resolution boundary provides the edge structure for a local Wheeler–DeWitt description. Because the diffeomorphism-invariant Hilbert space does not factorize, each diamond is equipped with a boundary-completed algebra ${\mathfrak{A}}_O$, ensuring the operational state ${\rho }_O$ and the semiclassical reference family ${\sigma }_O[\Lambda ]$ share identical operator content. Dynamics are posed as local statistical inference: the relative-entropy functional $S_{\mathrm{rel}}({\rho }_O\parallel {\sigma }_O[\Lambda ])$ quantifies the mismatch between data and reference. This yields the minimal operational axioms defining subsystems, intrinsic clocks, and regulated observables in a finite-resolution, background-independent setting. The topology-locked boundary capacity budget fixes an effective channel multiplicity $N\approx 1.23\times {10}^{11}$. Calibrating its coherent fraction to Newton’s constant determines a matching scale $M_s\approx 3.02\times {10}^{13}\mathrm{GeV}$. In the modular/KMS regime, the relative-entropy Hessian (Kubo–Mori metric) block-diagonalizes into orthogonal tensor, vector, and scalar response sectors. A heat-kernel expansion on the fixed $S^3\times S^1$ history manifold maps this near-equilibrium response to a matching-scale effective field theory, yielding the Einstein–Hilbert tensor structure, Yang–Mills susceptibilities, and leading mass deformations. Vector and scalar responses remain intensive, while the tensor response scales extensively with coherent channel multiplicity. The fixed modular protocol and quantized boundary currents imply ${\alpha }^{-1}\left(M_s\right)=4\pi k$ at integer levels k, while the reduced $R^2$ plateau sector yields linked cosmological targets: $n_s\simeq 0.965$, $r\simeq 0.0038$, and $A_s\simeq 2.1\times {10}^{-9}$. Translations between causal diamonds act as completely positive trace-preserving (CPTP) updates. The resulting open-modular Walsh filtration selects the three-dimensional degree-one sector as the algebraic basis for family structure. Treating continuum fields as the structured response of a finite boundary, the framework yields correlated, falsifiable relations for gravitational stiffness, gauge response, plateau cosmology, and threefold matter-sector organization from one minimal operational architecture. Full article

This paper presents a novel distributed coordination framework for multi-agent robotic swarms tailored for smart transportation applications. The proposed approach addresses the critical pre-transportation phase where a fleet of mobile robots, eventually with different sizes, must converge to fixed positions around an object to ensure effective caging within a user-defined prescribed time. By leveraging a time-varying diffeomorphic mapping based on an affine transformation, the strategy embeds prescribed-time guarantees within a swarm-inspired framework that maps agents between virtual and real reference frames. This methodology ensures the simultaneous achievement of precise target convergence, finite-time stability regardless of initial conditions, and inherent collision avoidance by explicitly considering the physical footprint of each robotic unit. The control protocol is first derived for scalar systems and subsequently extended to multidimensional robotic fleets using additional diffeomorphism-based techniques, which allow for the management of multiple non-interacting swarms to reduce network communication overhead. Full article

In freeform optical metrology, wavefront fitting over non-circular apertures is hindered by the loss of Zernike polynomial orthogonality and severe sampling grid distortion inherent in standard conformal mappings. To address the resulting numerical instability and fitting bias, we propose a unified framework curve-shortening flow (CSF)-guided progressive quasi-conformal mapping (CSF-QCM), which integrates geometric boundary evolution with topology-aware parameterization. CSF-QCM first smooths complex boundaries via curve-shortening flow, then solves a sparse Laplacian system for harmonic interior coordinates, thereby establishing a stable diffeomorphism between physical and canonical domains. For doubly connected apertures, it preserves topology by computing the conformal modulus via Dirichlet energy minimization and simultaneously mapping both boundaries. Benchmarked against state-of-the-art methods (e.g., Fornberg, Schwarz–Christoffel, and Ricci flow) on representative irregular apertures, CSF-QCM suppresses area distortion and restores discrete orthogonality of the Zernike basis, reducing the Gram matrix condition number from >900 to <8. This enables high-precision reconstruction with RMS residuals as low as $3\times {10}^{-3}\lambda$ and up to 92% lower fitting errors than baselines. The framework provides a unified, computationally efficient, and numerically stable solution for wavefront reconstruction in complex off-axis and freeform optical systems. Full article

When catastrophe theory is applied to the experimental modeling of catastrophe phenomena, it is impossible to know in advance the corresponding relationship and mapping form between the parameters of the actual catastrophe mathematical model and the parameters of the canonical catastrophe mathematical model. This gives rise to the problem in which the process of experimental modeling cannot be completed in many instances. To solve this problem, an experimental modeling method of catastrophe theory is proposed. It establishes the quantitative relationship between the actual catastrophe mathematical model and the canonical catastrophe mathematical model by assuming that the actual potential function is equal to the canonical potential function, and it uses a machine learning model to represent the diffeomorphism that can realize the error-free transformation of the two models. The method is applied to establish the experimental mathematical model of a cusp-type catastrophe for the Zeeman catastrophe machine. Through programming calculation, it is found that the prediction errors of the potential function, manifold, and bifurcation set of the established model are 0.0455%, 0.0465%, and 0.1252%, respectively. This indicates that the established model can quantitatively predict the catastrophe phenomenon. Full article

Poisson structures related to affine Courant-type algebroids are analyzed, including those related with cotangent bundles on Lie-group manifolds. Special attention is paid to Courant-type algebroids and their related R structures generated by suitably defined tensor mappings. Lie–Poisson brackets that are invariant with respect to the coadjoint action of the loop diffeomorphism group are created, and the related Courant-type algebroids are described. The corresponding integrable Hamiltonian flows generated by Casimir functionals and generalizing so-called heavenly-type differential systems describing diverse geometric structures of conformal type in finite dimensional Riemannian manifolds are described. Full article

This work proposes a unifying framework for extending PDE-constrained Large Deformation Diffeomorphic Metric Mapping (PDE-LDDMM) with the sum of squared differences (SSD) to PDE-LDDMM with different image similarity metrics. We focused on the two best-performing variants of PDE-LDDMM with the spatial and band-limited parameterizations of diffeomorphisms. We derived the equations for gradient-descent and Gauss–Newton–Krylov (GNK) optimization with Normalized Cross-Correlation (NCC), its local version (lNCC), Normalized Gradient Fields (NGFs), and Mutual Information (MI). PDE-LDDMM with GNK was successfully implemented for NCC and lNCC, substantially improving the registration results of SSD. For these metrics, GNK optimization outperformed gradient-descent. However, for NGFs, GNK optimization was not able to overpass the performance of gradient-descent. For MI, GNK optimization involved the product of huge dense matrices, requesting an unaffordable memory load. The extensive evaluation reported the band-limited version of PDE-LDDMM based on the deformation state equation with NCC and lNCC image similarities among the best performing PDE-LDDMM methods. In comparison with benchmark deep learning-based methods, our proposal reached or surpassed the accuracy of the best-performing models. In NIREP16, several configurations of PDE-LDDMM outperformed ANTS-lNCC, the best benchmark method. Although NGFs and MI usually underperformed the other metrics in our evaluation, these metrics showed potentially competitive results in a multimodal deformable experiment. We believe that our proposed image similarity extension over PDE-LDDMM will promote the use of physically meaningful diffeomorphisms in a wide variety of clinical applications depending on deformable image registration. Full article

In this paper, we find some new information on Legendrian dualities and extend them to the case of Legendrian dualities for continuous families of pseudo-spheres in general semi-Euclidean space. In particular, we construct all contact diffeomorphic mappings between the contact manifolds and display them in a table that contains all information about Legendrian dualities. Full article

In this paper, we study differential forms and vector fields on the orbit space of a proper action of a Lie group on a smooth manifold, defining them as multilinear maps on the generators of infinitesimal diffeomorphisms, respectively. This yields an intrinsic view of vector fields and differential forms on the orbit space. Full article

Brain science research often requires accurate localization and quantitative analysis of neuronal activity in different brain regions. The premise of related analysis is to determine the brain region of each site on the brain slice by referring to the Allen Reference Atlas (ARA), namely the regional localization of the brain slice. The image registration methodology can be used to solve the problem of regional localization. However, the conventional multi-modal image registration method is not satisfactory because of the complexity of modality between the brain slice and the ARA. Inspired by the idea that people can automatically ignore noise and establish correspondence based on key regions, we proposed a novel method known as the Joint Enhancement of Multimodal Information (JEMI) network, which is based on a symmetric encoder–decoder. In this way, the brain slice and the ARA are converted into a segmentation map with unified modality, which greatly reduces the difficulty of registration. Furthermore, combined with the diffeomorphic registration algorithm, the existing topological structure was preserved. The results indicate that, compared with the existing methods, the method proposed in this study can effectively overcome the influence of non-unified modal images and achieve accurate and rapid localization of the brain slice. Full article

Using the methods of symplectic geometry, we establish the existence of a canonical transformation from potential model Hamiltonians of standard form in a Euclidean space to an equivalent geometrical form on a manifold, where the corresponding motions are along geodesic curves. The advantage of this representation is that it admits the computation of geodesic deviation as a test for local stability, shown in recent previous studies to be a very effective criterion for the stability of the orbits generated by the potential model Hamiltonian. We describe here an algorithm for finding the generating function for the canonical transformation and describe some of the properties of this mapping under local diffeomorphisms. We give a convergence proof for this algorithm for the one-dimensional case, and provide a precise geometric formulation of geodesic deviation which relates the stability of the motion in the geometric form to that of the Hamiltonian standard form. We apply our methods to a simple one-dimensional harmonic oscillator and conclude with a discussion of the relation of bounded domains in the two representations for which Morse theory would be applicable. Full article

A tensor calculus adapted to the Anti-Newtonian limit of Einstein gravity is developed. The limit is defined in terms of a global conformal rescaling of the spatial metric. This enhances spacelike distances compared to timelike ones and in the limit effectively squeezes the lightcones to lines. Conventional tensors admit an analogous Anti-Newtonian limit, which however transforms according to a non-standard realization of the spacetime Diffeomorphism group. In addition to the type of the tensor the transformation law depends on, a set of integer-valued weights is needed to ensure the existence of a nontrivial limit. Examples are limiting counterparts of the metric, Einstein, and Riemann tensors. An adapted purely temporal notion of parallel transport is presented. By introducing a generalized Ehresmann connection and an associated orthonormal frame compatible with an invertible Carroll metric, the weight-dependent transformation laws can be mapped into a universal one that can be read off from the index structure. Utilizing this ‘decoupling map’ and a realization of the generalized Ehresmann connection in terms of scalar field, the limiting gravity theory can be endowed with an intrinsic Levi–Civita type notion of spatio-temporal parallel transport. Full article

In this work, we study the key role of generic Effective Field Theory (EFT) framework to quantify the correlation functions in a quasi de Sitter background for an arbitrary initial choice of the quantum vacuum state. We perform the computation in unitary gauge, in which we apply the Stückelberg trick in lowest dimensional EFT operators which are broken under time diffeomorphism. In particular, using this non-linear realization of broken time diffeomorphism and truncating the action by considering the contribution from two derivative terms in the metric, we compute the two-point and three-point correlations from scalar perturbations and two-point correlation from tensor perturbations to quantify the quantum fluctuations observed in the Cosmic Microwave Background (CMB) map. We also use equilateral limit and squeezed limit configurations for the scalar three-point correlations in Fourier space. To give future predictions from EFT setup and to check the consistency of our derived results for correlations, we use the results obtained from all classes of the canonical single-field and general single-field $P(X,\varphi )$ model. This analysis helps us to fix the coefficients of the relevant operators in EFT in terms of the slow-roll parameters and effective sound speed. Finally, using CMB observations from Planck we constrain all these coefficients of EFT operators for the single-field slow-roll inflationary paradigm. Full article

Anti-Newtonian expansions are introduced for scalar quantum field theories and classical gravity. They expand around a limiting theory that evolves only in time while the spatial points are dynamically decoupled. Higher orders of the expansion re-introduce spatial interactions and produce overlapping lightcones from the limiting isolated world line evolution. In scalar quantum field theories, the limiting system consists of copies of a self-interacting quantum mechanical system. In a spatially discretized setting, a nonlinear “graph transform” arises that produces an in principle exact solution of the Functional Renormalization Group for the Legendre effective action. The quantum mechanical input data can be prepared from its $1+0$ dimensional counterpart. In Einstein gravity, the anti-Newtonian limit has no dynamical spatial gradients, yet remains fully diffeomorphism invariant and propagates the original number of degrees of freedom. A canonical transformation (trivialization map) is constructed, in powers of a fractional inverse of Newton’s constant, that maps the ADM action into its anti-Newtonian limit. We outline the prospects of an associated trivializing flow in the quantum theory. Full article