Search Results (141)

Mean curvature flow (MCF) is an effective tool for investigating the geometry and topology of submanifolds under suitable curvature pinching conditions. While most existing convergence results in complex projective spaces rely on pointwise curvature assumptions, much less is known about the case of integral curvature constraints. In this paper, we consider the MCF of smooth closed submanifolds of small codimension immersed in ${\mathbb{CP}}^{{\displaystyle \frac{n+k}{2}}}$. We establish our main theorem under an explicit integral curvature pinching condition: the $L^p$-norm of the second fundamental form of the initial submanifold is bounded above by a small constant depending only on the dimension and the exponent p. The proofs are based on evolution equations for geometric quantities, Sobolev inequalities, and Moser iteration, which together yield uniform curvature estimates and preserve the required integral pinching condition along the flow. These estimates allow us to reduce the problem to previously established convergence criteria under pointwise curvature pinching conditions. Consequently, the MCF either shrinks to a round point in finite time or converges smoothly to a totally geodesic submanifold as $t\to \infty$. As a consequence, we obtain a differentiable sphere theorem: any such submanifold satisfying the same integral curvature pinching condition is diffeomorphic to either the standard sphere ${\mathbb{S}}^n$ or the complex projective space ${\mathbb{CP}}^{{\displaystyle \frac{n}{2}}}$. Full article

Non-rigid image registration is a fundamental problem in medical imaging and a representative example of continuous transformation modeling in image processing. Diffeomorphic registration methods, such as Large Deformation Diffeomorphic Metric Mapping (LDDMM) and its PDE-constrained variants (PDE-LDDMM), provide mathematically grounded formulations with strong geometric guarantees for transformation quality. However, existing approaches face persistent trade-offs between numerical stability, accuracy, and computational efficiency. Recent work has explored implicit neural representations (INRs) and neural ordinary differential equations (NODEs) as flexible neural representations for modeling continuous transformations. Despite their increasing adoption, their practical behavior and limitations in diffeomorphic registration remain insufficiently understood. In this paper, we present a unified formulation of INR- and NODE-based registration methods within LDDMM and PDE-LDDMM, enabling a systematic and controlled comparison across architectures, sampling strategies, and numerical solvers. Our analysis reveals fundamental trade-offs between these approaches. In particular, we show that MLP-based INR formulations introduce significant computational overhead and rely on sampling strategies that can degrade smoothness and lead to the increased occurrence of non-diffeomorphic transformations at higher resolutions. Moreover, these approximations do not fully alleviate the computational cost, with some variants exceeding the costs of expensive classical optimization-based methods. In contrast, NODE-based formulations and downsampling strategies consistently provide transformations with more controlled Jacobian extrema while maintaining competitive computational performance. Among the evaluated methods, the original NODE-LDDMM and NODE-PDE-LDDMM formulations achieve the most favorable trade-offs between registration accuracy, geometric consistency, and computational efficiency. These findings provide clear insights into the design of neural representations for continuous transformation modeling, with practical implications for diffeomorphic registration and computational anatomy applications. Full article

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

This paper presents a decentralized adaptive generalized-minimum-variance (GMV) control framework for large-scale stochastic nonlinear systems composed of interconnected multi-input multi-output (MIMO) Hammerstein subsystems with unknown time-varying parameters. Each subsystem consists of a coupled multivariable static nonlinearity represented on a known invertible basis, followed by a matrix-polynomial dynamic block affected by colored noise and delayed input–output interconnections. The proposed scheme estimates only identifiable composite Hammerstein parameters through a decentralized recursive extended least-squares algorithm with forgetting, thereby avoiding the non-unique separation of nonlinear and linear gains. A constructive matrix Diophantine identity is established to derive an optimal multi-step predictor, leading to a GMV control law expressed as a multivariable polynomial equation in the current input. Sufficient conditions for real solvability, mean-square boundedness, and near-optimal adaptive tracking are provided using Hadamard–Lévy global-diffeomorphism, minimum-phase, small-gain, persistent-excitation, strict-positive-realness, and convex-projection arguments, and the implemented controller—inexact Newton solver with fallback and persistent dither—is itself covered by the analysis. The analysis further shows that delayed interconnections become measurable and can be exactly compensated, while robustness to basis under-modeling is explicitly quantified. Simulation results on an interconnected two-subsystem MIMO Hammerstein process with coupled cubic nonlinearities, colored noise, delayed interactions, and time-varying parameters—run in the forgetting-factor regime required by the theory, with measured persistent excitation and complete solver diagnostics—demonstrate operational-noise-floor tracking and a $2.3$-fold mean-RMSE reduction relative to the strongest linear-MIMO surrogate, while a channel-wise SISO Hammerstein design fails structurally and a feedback-linearization controller with exactly known nonlinearity offers no advantage. The study further demonstrates scalability on a chain of four subsystems with size-independent per-subsystem computational cost, validates a physically motivated interconnected coupled-tank network with progressive-valve nonlinearities, and confirms agreement between the observed stability limits and the predicted small-gain boundary. Full article

This paper investigates scaling symmetry in thermodynamics by unifying constrained Hamiltonian dynamics with symplectic and contact geometries. Through the mathematical processes of contactization and symplectization, we demonstrate that fixing an extended global scale variable effectively recovers the standard thermodynamic description in terms of scale-invariant quantities. The geometric formalism is illustrated by establishing the diffeomorphism between the Lagrangian submanifolds of ideal and van der Waals gases. Finally, applying this framework to a Schwarzschild black hole reveals that changing the scaling weights of entropy and internal energy is a fundamental physical requirement to accommodate non-isothermal dynamics. Full article

Transformer-based deformable brain MRI registration achieves high overlap accuracy, but predicted displacement fields can contain voxels with a non-positive Jacobian determinant—local foldings that violate the diffeomorphism assumption required by tensor-based morphometry and atlas-fusion segmentation workflows. We introduce HypEReg, a non-linear hyperelastic regularizer that acts directly on the Jacobian determinant of the predicted displacement field. HypEReg couples a clamped-rational volume-distortion penalty ${(det{J}_{\varphi }-1)}^2/max(det{J}_{\varphi },ϵ)$ with an explicit per-voxel anti-folding hinge ${[max(0,ϵ-det{J}_{\varphi })]}^2$, integrated as a purely loss-side module into a TransMorph backbone with no inference-graph modifications. On the IXI atlas-to-subject benchmark (115 test subjects), HypEReg-TransMorph maintains grouped Dice (0.7537) while reducing the $det\left(J_{\varphi }\right)\le 0$ voxel ratio from $1.502\times {10}^{-2}$ (TransMorph) to $1.5\times {10}^{-5}$, with identical per-case runtime and parameter count to the unregularized baseline. In strict zero-shot transfer to OASIS Learn2Reg test pairs (no fine-tuning), HypEReg-TransMorph achieves Dice 0.7756 with a $det\left(J_{\varphi }\right)\le 0$ ratio of $7.6\times {10}^{-5}$, roughly two orders of magnitude below plain TransMorph zero-shot (Dice 0.7691; ratio $9.6\times {10}^{-3}$); downstream multi-atlas label fusion further confirms the practical benefit of fold suppression (fused Dice 0.8271 vs. 0.8201 for TransMorph). OASIS-2 longitudinal and ROI analyses support deformation plausibility (lower folding/SDlogJ and stronger ventricular ROI agreement), while clinical-covariate associations remain exploratory rather than biomarker-validating. Determinant-level, non-linear hyperelastic regularization substantially suppresses folding in Transformer dense-flow brain MRI registration while preserving alignment accuracy and adding zero inference cost, providing a practical drop-in regularization strategy that improves the reliability of deformation fields for morphometry-oriented deformable registration. 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

Every stable 4-sphere is identified with the double branched covering space of a trivial surface-knot space. As a result of Wall, it is known that any two orthogonal bases of every stable 4-sphere are transformed into each other by an orientation-preserving diffeomorphism of the stable 4-sphere. In this paper another proof of Wall’s result is presented, and strengthened in the sense that the lift of an equivalence of the trivial surface-knot space can be taken as the diffeomorphism. Two applications are made. The first shows that every orientation-preserving diffeomorphism of every stable 4-sphere is nothing but the double branched covering lift of an equivalence of a trivial surface-knot space up to a smooth isotopy and a composition with an identity-shift. The second gives a similar result for TOP stable 4-spheres. Here, even if it is a smooth 4-manifold, unless it is diffeomorphic to the stable 4-sphere, the TOP trivial surface-knot space cannot be smooth. Full article

Background/Objectives: The posterior commissure (PC) and habenular commissure (HC) are fine-caliber midline fiber bundles located within the epithalamic roof of the third ventricle. Their small size and deep anatomical position render them vulnerable during fiber dissection, and it is difficult to delineate reliably with conventional diffusion MRI. To define a reproducible microsurgical strategy for three-dimensional exposure of the PC and HC, and to evaluate their tractographic representation using high-resolution diffusion template data. Methods: Four formalin-fixed adult cadaveric brains were prepared using a modified Klingler technique and underwent systematic microsurgical fiber dissection focused on preservation of the epithalamic roof and midline commissures. Diffusion MRI data from the Human Connectome Project (HCP-1065) were reconstructed in MNI space using q-space diffeomorphic reconstruction in DSI Studio to attempt deterministic tractographic reconstruction of the PC and HC. Results: In all specimens, the PC was identified as a compact transverse bundle superior to the rostral cerebral aqueduct within the inferior pineal lamina. The HC appeared as a thinner band superior to the pineal recess, interconnecting the bilateral habenular nuclei and separated from the PC by the hypothalamic sulcus. A midline-prioritized dissection sequence facilitated preservation of commissural continuity. Deterministic tractography reproduced adjacent peduncular trajectories but failed to consistently reconstruct discrete HC or PC streamlines. Conclusions: Cadaveric fiber dissection remains the most reliable method for studying the fine commissural anatomy of the epithalamus. A midline-first, roof-preserving strategy enhances visualization of the PC and HC and may have implications for posterior third ventricular surgery and stereotactic targeting. 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

This paper presents a detailed re-examination of the conformalcompactification $\overline{\mathbb{M}}$ of Minkowski space $\mathbb{M}$, constructed as the projective null cone of the six-dimensional space ${\mathbb{R}}^{4,2}$. We provide an explicit and basis-independent formulation, emphasizing geometric clarity. A central result is the explicit identification of $\overline{\mathbb{M}}$ with the unitary group $U\left(2\right)$ via a diffeomorphism, offering a clear matrix representation for points in the compactified space. We then systematically construct and analyze the action of the full conformal group $\mathrm{O}(4,2)$ and its connected component ${\mathrm{SO}}_0(4,2)$ on this manifold. A key contribution is the detailed study of the double cover, $\tilde{\mathbb{M}}$, which is shown to be diffeomorphic to $S^3\times S^1$. This construction resolves the non-effectiveness of the $\mathrm{SO}(4,2)$ action on $\overline{\mathbb{M}}$, yielding an effective group action on the covering space. A significant portion of our analysis is devoted to a precise and novel geometric characterization of the conformal infinity. Moving beyond the often-misrepresented “double cone” description, we demonstrate that the infinity of the double cover, ${\tilde{\mathbb{M}}}_{\infty }$, is a squeezed torus (specifically, a horn cyclide), while the simple infinity, ${\overline{\mathbb{M}}}_{\infty }$, is a needle cyclide. We provide explicit parametrizations and graphical representations of these structures. Finally, we explore the embedding of five-dimensional constant-curvature spaces, whose boundary is the compactified Minkowski space. The paper aims to clarify long-standing misconceptions in the literature and provides a robust, coordinate-free geometric foundation for conformal compactification, with potential implications for cosmology and conformal field theory. Full article

Ultrasound images have some limitations, such as low signal-to-noise ratio (SNR), speckle noise, lower dynamic range, blurred boundaries, and shadowing; therefore, ultrasound image registration is an important task for estimating tissue motion and analyzing tissue mechanical properties. In this paper, an effective non-rigid ultrasound image registration method is proposed. By integrating intensity, local phase information, and descriptor matching under a variational framework, we can find and track the non-rigid transformation of each pixel under diffeomorphism between the source and target images based on the warping technique. Experiments using simulation and in vivo ultrasound images of the human carotid artery are conducted to demonstrate the advantages of the proposed algorithm, which will act as an important supplement to current ultrasound image registration. Full article

Classic machine learning and regime identification methods applied to financial time series lack theoretical guarantees and exhibit systematic failure modes: heavy-tails invalidate moment-based geometry, rendering distances and centroids dominated by extremes or unstable; jumps violate smoothness, destabilizing local regressions, kernel methods, and gradient-based learning; and non-stationarity disrupts neighborhood relations, so distances in classical feature spaces no longer reflect meaningful proximity. To address these challenges, we propose a topology-based machine-learning framework grounded on probabilistic reconstruction of state-space geometry, which replaces moment- and smoothness-dependent representations with deformation-stable summaries of state-space geometry, preserving neighborhoods, adjacency, and topology. The finite-sample validity of homeomorphic state-space reconstruction, required for topology-based machine learning, is assessed through numerical studies on synthetic data with heavy tails, jumps, and known ground-truth regimes. Further diagnostics of local invertibility and bounded geometric distortion quantify when embedding windows are consistent with local diffeomorphic behavior, enabling metric-sensitive, geometry-aware learning. Clustering of Hilbert-space summaries accurately recovers underlying market tail-risk regimes with robust results across selected filtrations. Temporal, feature-space, and cluster-label null tests confirm that topology-based clustering captures genuine topological structure rather than noise or artifacts, and encodes temporal dependencies at local, mesoscopic, and network levels associated with market regimes. Full article

This paper investigates the finite-horizon survival probability for a system of correlated arithmetic Brownian motions with heterogeneous drifts and volatilities, focusing on the event in which one component remains strictly below all others. Using a whitening transformation of the covariance structure, we reduce the problem to the survival of a standard Brownian motion in a simplicial cone, characterized by its spherical cross-section. While explicit solutions are available in low dimensions, we address the computationally challenging tetrahedral angular case. We derive a semi-analytic formula for the survival probability via an eigenfunction expansion of the Dirichlet Laplace–Beltrami operator on this curved domain. For efficient implementation, we construct a diffeomorphism from the spherical tetrahedron to a fixed Euclidean tetrahedron, enabling the computation of angular eigenpairs through a stable finite-element scheme. For higher-dimensional regimes, we also introduce a covariance-based difficulty index and geometric bounds based on an inscribed spherical cap to assess spectral convergence and estimate long-time decay rates. Numerical experiments show that this offline–online approach achieves high accuracy and substantial speedups relative to Monte Carlo benchmarks. Full article