Geometrically characterizes virtual nonlinear nonholonomic constraints using symplectic methods.
problem Characterizing virtual nonlinear nonholonomic constraints geometrically.
method Geometric characterization using symplectic structures and Chetaev equations.
result A unique control law exists to satisfy virtual constraints, and closed-loop dynamics are projections of uncontrolled dynamics.
New unknots with geometric constraints exist, proving a long-standing conjecture.
problem Existence of distinct isotopy classes of physical unknots with geometric constraints.
method Parametrised thickness and geometric thresholds to fragment isotopy classes.
result Existence of gordian unknots with prescribed geometric constraints.
Geometric constraints help classify hyperbolic polytopes.
problem Classifying reflective anisotropic Lorentzian lattices and cocompact arithmetic hyperbolic reflection groups.
method Established geometric constraints on compact Coxeter polytopes in hyperbolic spaces.
result Geometric constraints are useful for classifying hyperbolic polytopes.
We formalize geometrically the idea that the (de Donder) Hamiltonian formulation of a higher derivative Lagrangian field theory can be constructed understanding the latter as a first derivative theory subjected to constraints.
Study of motion constraints and path-following on 3D space.
problem Path-following with non-holonomic constraints on R3. method Exploration of geometric structure and construction of guiding vector fields.
result General principles for constructing guiding vector fields for path-following.
Geometric theory explains substitutability in market outcomes based on production constraints.
problem Understanding substitutability in markets with structured feasible products.
method Modeling the set of feasible products as a compact Riemannian manifold to study intrinsic geometry and its effects on substitutability.
result Intrinsic geometry of the feasible set governs substitutability and market outcomes, with curvature controlling technological substitution elasticity.
This paper studies Hamilton-Jacobi equations for magnetic systems with constraints.
problem Understanding dynamics of magnetic systems with geometric constraints.
method Developed Hamilton-Jacobi equations for magnetic systems with nonholonomic constraints.
result Revealed relationships between magnetic structures, constraints, and dynamics.
Study on harmonic maps from surfaces to homogeneous spaces, focusing on bubble formation and geometric constraints.
problem Understanding the behavior of harmonic maps from surfaces to homogeneous spaces, especially in the presence of bubbles.
method Refined asymptotic expansions and obstruction relations for sequences developing a single bubble, geometric constraints for weakly conformal maps.
result New geometric constraints on the tangent planes of the limit map and bubble, depending on the dimensionality.
We treat the vakonomic dynamics with general constraints within a new geometric framework which will be appropriate to study optimal control problems. We compare our formulation with Vershik-Gershkovich one in the case of linear constraints. We show how nonholonomic mechanics also admits a new geometrical description w…
The paper explores solving inverse problems for ODEs with and without constraints.
problem Understanding when second order ODEs can represent Lagrangian models with or without constraints.
method Geometric techniques to address the inverse problem for both constrained and unconstrained systems of second order ODEs.
result The constrained case presents more ambiguities and complexities than the unconstrained one.
Study on packing links with geometric constraints.
problem Maximizing link density in space with geometric restrictions.
method Investigates packing essential links within Euclidean space.
result Upper bounds on maximal density are found, but are large.
A new geometrical setting for classical field theories is introduced. This description is strongly inspired in the one due to Skinner and Rusk for singular lagrangians systems. For a singular field theory a constraint algorithm is developed that gives a final constraint submanifold where a well-defined dynamics exists.…
In the last two decades, significant effort has been put in understanding and designing so-called structure-preserving numerical methods for the simulation of mechanical systems. Geometric integrators attempt to preserve the geometry associated to the original system as much as possible, such as the structure of the co…
Method optimizes knotting pathways in constrained polymers.
problem Understanding how geometric constraints affect knot formation in polymers.
method Topological steering using knotoid spectrum and mean unravelling number.
result Geometric constraints increase the frequency of twist knots in polymers.
New metrics improve landing algorithms for orthogonality constraints.
problem Optimizing landing algorithms with orthogonality constraints.
method Proposed a family of metrics over full-rank matrices to enhance landing algorithms.
result Natural extension of β-metric improves landing performance.
Estimates KVol on surfaces with geometric constraints.
problem Determining the intersection of closed curves on translation surfaces.
method Geometric constraints on angles and indentifications of sides.
result Sharp estimate for KVol on Bouw-Möller surfaces with a unique singularity.
SketchGraphs dataset aids in modeling CAD designs.
problem Training models to reason about CAD designs efficiently.
method Collection of 15 million sketches with geometric constraint graphs.
result Demonstrated use cases for generative modeling and conditional generation.
Reduces Lie (bi-)algebroids and Dirac manifolds using constraint vector bundles.
problem Reduction of Lie (bi-)algebroids and Dirac manifolds.
method Introduces constraint manifolds and constraint vector bundles; proves constraint Serre-Swan theorem; introduces Cartan calculus for constraint forms and multivector fields; shows compatibility with reduction.
result Reduction procedure for Lie (bi-)algebroids and Dirac manifolds.
Optimization with inequality constraints using embedded gradient vector field method
problem Optimization with inequality constraints
method Geometric framework using quadratic slack variables
result Derives Lagrange multiplier functions and second-order optimality conditions
New constraints rule out some optimal domains for helicity maximisation.
problem Finding a smooth domain of fixed volume that maximizes helicity.
method Established additional geometric constraints on optimal domains.
result Ruled out the optimality of a broad class of solid tori.
LightGCNet simplifies AI for soft sensors, reducing complexity and training time.
problem Complex and resource-intensive deep learning models for soft sensors.
method LightGCNet uses compact angle constraints and node pool strategy for efficient learning.
result LightGCNet achieves small network size, fast learning, and good generalization.
A multilevel optimization method for constrained problems.
problem Regularized constrained linear inverse problems with box constraints.
method Geometric multilevel optimization with varying discretization levels.
result Preserves feasibility of updates while speeding up computations.
Study optimal consumption with relaxed benchmarks and drawdown constraints.
problem Optimal consumption under relaxed benchmark tracking and consumption drawdown constraint.
method Transformed stochastic control problem into regular control problem with state-control constraints, then solved using dual transform and optimal consumption behavior.
result Closed-form solution for optimal investment and consumption in feedback form.
Introduces GFC for learning complex dynamical systems with geometric constraints.
problem Challenges in accurately modeling and predicting complex dynamical systems with geometric constraints.
method Geometric Contact Flows (GFC) using Riemannian and Contact geometry as inductive biases.
result Ensemble of contactomorphisms adapt the latent contact Hamiltonian model to target dynamics while preserving desirable properties.
Convex regression is a promising area for bridging statistical estimation and deterministic convex optimization. New piecewise linear convex regression methods are fast and scalable, but can have instability when used to approximate constraints or objective functions for optimization. Ensemble methods, like bagging, sm…
This paper focuses on martingale optimal transport problems when the martingales are assumed to have bounded quadratic variation. First, we give a result that characterizes the existence of a probability measure satisfying some convex transport constraints in addition to having given initial and terminal marginals. Sev…
Paper constructs observables using multisymplectic geometry and algebraic methods.
problem Building observables in multisymplectic geometry.
method Uses L∞-algebras, Gerstenhaber algebras, BV-modules, and constraint triples. result Reconstructs and explains recent geometric results.
Critical trajectories in a sphere are found for a specific bending functional.
problem Finding closed trajectories in a sphere for a specific bending functional.
method Existence of infinitely many closed trajectories shown for a given Lagrange multiplier.
result Existence of closed trajectories dependent on a pair of relatively prime natural numbers.
New constraints found for algebro-geometric subgroups of mapping class groups.
problem Constraints for algebro-geometric subgroups of mapping class groups.
method Using deep work of Gibney, Keel, and Morrison, constraints on the Shafarevich morphism are derived to prove the infinite restriction of certain representations.
result Most Reshetikhin-Turaev representations of the mapping class group restrict to infinite representations on algebro-geometric subgroups when the genus is at least 3.
Unified treatment of stability problems in geometry and analysis.
problem Spherical closeness of hypersurfaces under geometric constraints.
method Estimate relating distance to geodesic spheres with norms of traceless Hessian operator.
result Unified treatment of stability problems in geometry and analysis.
DMT enhances deep neural networks to better preserve data structures.
problem Preserving geometric, topological, and distributional structures of data in NLDR.
method Deep manifold transformation (DMT) using cross-layer LGP constraints.
result DMT networks outperform existing NLDR methods in preserving data structures.
New method learns dynamics from sparse data using geometric constraints.
problem Learning dynamics from sparse, undersampled data.
method Reformulates inference as a stochastic control problem, using geometry-driven path augmentation.
result Accurately recovers stochastic dynamics from extremely undersampled data.
Intermediate logic of all convex polyhedra is axiomatized.
problem Defining and axiomatizing intermediate logic for convex polyhedra.
method Using Jankov-Fine formulas, classical polyhedral geometry, and p-morphic images to establish completeness.
result A finite axiomatisation of PL for all convex polyhedra.
Study magnetic Hamiltonian systems with constraints, deriving Hamilton-Jacobi equations.
problem Understanding dynamics of controlled magnetic Hamiltonian systems with constraints.
method Defined CMH system, derived Hamilton-Jacobi equations for different constraints.
result Invariant solutions of Hamilton-Jacobi equations under CMH-equivalence.
Paper finds optimal selling rule for pairs trading with stock constraints.
problem Identifying the best time to sell in pairs trading of stocks.
method Optimal pairs-trading selling rule with constraints on trading.
result Closed-form solution for optimal policy determined by a threshold curve.
Optimal control problems on Riemannian manifolds are solved by penalizing constraint violations.
problem Optimal control problems with velocity constraints on Riemannian manifolds.
method Penalizing constraint violations and showing convergence to hard-constrained solutions.
result Solutions to soft-constrained problems converge to solutions of hard-constrained problems as penalty parameter increases.
Adversarial examples are a pervasive phenomenon of machine learning models where seemingly imperceptible perturbations to the input lead to misclassifications for otherwise statistically accurate models. We propose a geometric framework, drawing on tools from the manifold reconstruction literature, to analyze the high-…
VRSGT algorithm reduces orthogonality constraints in decentralized optimization.
problem Decentralized optimization with orthogonality constraints.
method VRSGT algorithm with variance reduction and orthogonal techniques.
result VRSGT achieves convergence rate of O(1 / k) for orthogonality constraints.
This paper studies nonholonomic constraints in Hamiltonian systems, deriving equations and theorems.
problem Analyzing nonholonomic constraints in Hamiltonian systems.
method Deriving distributional RCH systems, geometric constraint conditions, and Hamilton-Jacobi theorems.
result Derives precise geometric constraint conditions and Hamilton-Jacobi theorems for nonholonomic systems.
We investigate geometric aspects of double field theory (DFT) and its formulation as a doubled membrane sigma-model. Starting from the standard Courant algebroid over the phase space of an open membrane, we determine a splitting and a projection to a subbundle that sends the Courant algebroid operations to the correspo…
We address the restriction problem for viscosity subsolutions of a fully nonlinear PDE on a manifold Z. The constraints on the restrictions of smooth subsolutions to a submanifold X in Z determine a restricted subequation on X. The problem is to show that general (upper semi-continuous) subsolutions restrict to satisfy…
Proof of local well-posedness for a specific boundary condition in general relativity.
problem Initial boundary value problem in general relativity with umbilic boundary condition.
method Wave coordinates and key observation of momentum constraint validity for umbilic boundaries.
result Local well-posedness established for the initial boundary value problem.
New friction model for geometric locomotion systems.
problem Modeling asymmetric friction in locomotion systems.
method Introducing asymmetric friction into geometric locomotion models using Finsler metrics.
result Generalized motility map for systems with asymmetric friction.
The paper explores geometric relationships in manifolds with curvature constraints, proving new inequalities and rigidity results.
problem Understanding geometric features of manifolds with curvature constraints.
method Comparison theorems and spacetime harmonic functions.
result Partial resolution of Gromov's conjecture and new characterizations of geometries.
New boundary and point constraints for controlling conformal surfaces.
problem Controlling the geometry of surfaces defined by minimizers of conformal variational problems.
method Introducing new boundary conditions, point constraints, and flux constraints to control the metric and conformal scale factor.
result Introduces intuitive controls for exploring a subspace of conformal immersions.
Proves strict inequality for minimizers of Willmore energy under isoperimetric constraints.
problem Minimizing the Willmore energy under isoperimetric constraints.
method Connected sum approach, building on previous work by Keller-Mondino-Rivière.
result Existence of minimizers for the isoperimetric constrained Willmore problem in every genus.
The paper introduces a new divergence for portfolio management to outperform a benchmark.
problem Maximizing expected utility of outperformance over a benchmark with constraints.
method Uses α-Bregman-Wasserstein divergence to penalize underperformance more than overperformance. result Proves existence and uniqueness of optimal portfolio strategy and conditions for constraints binding.
The paper establishes pressure gaps for manifolds with flat subtori singularities.
problem Understanding phase transitions in nonpositively curved manifolds with flat subtori.
method Derives a pressure gap criterion for closed rank 1 manifolds with specific singular sets and proves Hölder continuity of geometric potentials.
result Geometric potentials have pressure gaps and no phase transitions under certain curvature constraints.