Abstract not provided enough details, focusing on vector bundles and orbits.
problem Understanding continuous representations of semisimple Lie groups.
method Not specified in the abstract.
result Not specified in the abstract.
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48 results for “continuous adjoint”
Discrete Adjoint Matchingstat.ML
Proposes DAM for optimizing discrete generative models.
problem Challenges in optimizing discrete generative models.
method Discrete Adjoint Matching (DAM) for discrete state spaces.
result Demonstrates effectiveness on synthetic and mathematical reasoning tasks.
QAM uses adjoint matching to optimize continuous-action RL policies efficiently.
problem Efficient optimization of expressive diffusion or flow-matching policies with respect to a Q-function.
method QAM leverages adjoint matching to bypass the numerical instability of backpropagation through multi-step denoising processes.
result QAM consistently outperforms prior approaches on hard, sparse reward tasks in offline and offline-to-online RL.
AdjointDEIS simplifies diffusion model optimization.
problem Optimizing diffusion models with respect to a differentiable metric.
method Novel bespoke ODE solvers for continuous adjoint equations.
result Continuous adjoint equations simplify to a simple ODE, improving efficiency.
Graph continuous operators become Riesz continuous after multiplication by unitary operators.
problem Characterizing Riesz continuity of graph continuous operators.
method Multiplication by unitary operators to transform graph continuity to Riesz continuity.
result The index of graph continuous families of Fredholm operators coincides with N. Ivanov's index.
Unified framework extends adjoint Schrödinger bridge sampler to discrete spaces.
problem Challenges in learning discrete neural samplers due to gradients and combinatorial complexity.
method Introduces discrete ASBS, a unified framework that extends adjoint Schrödinger bridge sampler to discrete spaces.
result Empirically, discrete ASBS achieves competitive sample quality with significant advantages in training efficiency and scalability.
We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous fa…
Study limits of adjoint orbits for Lie groups, describing nilpotent orbits.
problem Understanding limits of adjoint orbits for Lie groups.
method Systematic and topological study of limits of continuous families of adjoint orbits for non-compact simple Lie groups.
result Explicit description of nilpotent orbits in terms of Richardson orbits for hyperbolic semisimple elements.
The generator gradient estimator is an adjoint state method for stochastic differential equationsmath.OC
New estimator for SDEs is shown to be an adjoint state method.
problem Estimating gradients for overparameterized SDEs efficiently.
method Demonstrates generator gradient estimator as an adjoint state method.
result Generator gradient estimator is an adjoint state method for SDEs.
We prove a theorem relating the automorphism group of a Cartan geometry to the group on which the geometry is modeled: a component of the adjoint representation of the first embeds in the adjoint representation of the second. Consequences of the theorem include general bounds on the rank and nilpotence degree of an aut…
In this article we prove a generalization of Weyl's criterion for the spectrum of a self-adjoint nonnegative operator on a Hilbert space. We will apply this new criterion in combination with Cheeger-Fukaya-Gromov and Cheeger-Colding theory to study the -form essential spectrum over a complete manifold with vanishing…
Investigates the relationship between ResNets and Neural ODEs, quantifying their closeness and providing training methods.
problem Quantifying the distance between ResNet dynamics and Neural ODE solutions.
method Bounding the distance between hidden state trajectories and Neural ODE solutions, using gradient descent and Heun's method.
result Gradient descent and Heun's method can implicitly regularize ResNets towards Neural ODEs, especially for smooth residual functions.
Graph neural networks learn PDEs from sparse, irregular data.
problem Learning PDEs from irregularly spaced data.
method Continuous-time differential model with graph neural networks for arbitrary discretizations.
result Efficient inference with continuous-time adjoint method.
For a -algebra of compact operators and a compact manifold we prove that the Hodge theory holds for -elliptic complexes of pseudodifferential operators acting on smooth sections of finitely generated projective -Hilbert bundles over For these -algebras, we get also a topological isomorphis…
Study on removing sets and uniqueness of diffusion operators on various spaces.
problem Determining the effect of removing small sets on the self-adjointness and uniqueness of diffusion operators.
method Analyzes symmetric diffusion operators on metric measure spaces, proving a truncation result for potentials.
result Characterizes the critical size of removed sets and their effect on operator properties.
Let D be a self-adjoint differential operator of Dirac type acting on sections in a vector bundle over a closed Riemannian manifold M. Let H be a closed D-invariant subspace of the Hilbert space of square integrable sections. Suppose D restricted to H is semibounded. We show that every element u in H has the weak uniqu…
We study realizations of pseudodifferential operators acting on sections of vector-bundles on a smooth, compact manifold with boundary, subject to conditions of Atiyah-Patodi-Singer type. Ellipticity and Fredholm property, compositions, adjoints and self-adjointness of such realizations are discussed. We construct regu…
We continue the investigation of the correspondence between systems of conservation laws and congruences of lines in projective space. Relationship between "additional" conservation laws and hypersurfaces conjugate to a congruence is established. This construction allows us to introduce, in a purely geometric way, the …
We establish new Calderón reproducing formulas for self-adjoint operators that generate strongly continuous groups with finite propagation speed. These formulas allow the analysing function to interact with through holomorphic functional calculus whilst the synthesising function interacts with through funct…
This work presents a partitioned solution procedure to compute shape gradients in fluid-structure interaction (FSI) using black-box adjoint solvers. Special attention is paid to project the gradients onto the undeformed configuration. This is due to the mixed Lagrangian-Eulerian formulation of large-displacement FSI in…
In this paper we continue the study of spectral properties of Laplacians associated with an arbitrary smooth distribution on a compact manifold, initiated in a previous paper. Under assumption that the singular foliation generated by the distribution is smooth, we prove that the Laplacian associated with the distributi…
Let M be a connected Riemannian manifold and let D be a Dirac type operator acting on smooth compactly supported sections in a Hermitian vector bundle over M. Suppose D has a self-adjoint extension A in the Hilbert space of square-integrable sections. We show that any -section contained in a closed A-invariant…
Derives adjoint polynomials of torus knots in explicit form.
problem Understanding adjoint invariants of torus knots.
method Closed-form double sum expression derivation.
result Explicit double sum form of adjoint polynomials.
A hermitian algebra is a unital associative -algebra endowed with an involution such that the spectra of self-adjoint elements are contained in . In the case of an algebra endowed with a Mackey-complete, locally convex topology such that the set of invertible elements is open an…
Spectral Sectionsmath.SP
The paper generalizes spectral section concepts to non-compact spaces.
problem Generalizing spectral sections to non-compact base spaces.
method Generalization to arbitrary base spaces, applications to cobordism theorems, investigation of Riesz continuity.
result If a family of operators has a spectral section, it is Riesz continuous.
We describe a canonical form for linear differential operators that are formally self-adjoint or formally skew-adjoint.
Proves formal self-adjointness of certain differential operators.
problem Verifying conjectures about differential operators.
method Proving formal self-adjointness through mathematical proof.
result Proves two conjectures about differential operators.
Study of adjoint orbits in simplest non-trivial Lie algebra case.
problem Geometric properties of adjoint orbits in .
method Analysis of adjoint orbits, showing three possibilities: hyperboloids or cones.
result Just three possibilities for adjoint orbits: hyperboloids or cones.
Abstracts a construction of boundary triplets for self-adjoint elliptic problems.
problem Computing the index of families of self-adjoint elliptic boundary problems.
method Abstract axiomatic version of boundary triplets and their applications.
result Analytic proof of index theorem and computation of index differences.
Prove integrality of genus- indices with adjoint Reidemeister torsions for twist knots and meridians.
problem Prove integrality of genus- indices with adjoint Reidemeister torsions for twist knots and meridians.
method Consider the sum of the adjoint Reidemeister torsions and prove integrality for twist knots and meridians.
result Prove integrality of genus- indices with adjoint Reidemeister torsions for twist knots and meridians.
Inexact bilevel stochastic gradient methods for constrained and unconstrained lower-level problemsmath.OC
New methods solve complex optimization problems in machine learning.
problem Challenges in stochastic bilevel optimization with constraints and high variables.
method Inexact bilevel stochastic gradient methods for constrained and unconstrained lower-level problems.
result Comprehensive convergence theory for both unconstrained and constrained cases.
Quantizes Stäckel integrable systems into self-adjoint operators.
problem Quantizing Stäckel integrable systems into self-adjoint operators.
method Constructs commutative self-adjoint operators from quadratic Hamiltonians in involution.
result Proves multiplicative separation of variables for Stäckel integrable systems.
We give explicit descriptions of the adjoint group of the Coxeter quandle associated with an arbitrary Coxeter group . The adjoint group of turns out to be an intermediate group between and the corresponding Artin group , and fits into a central extension of by a finitely generated free abel…
SONODEs and ANODEs improve learning of second order dynamics.
problem Learning dynamics governed by second order laws.
method Extended adjoint sensitivity method and theoretical analysis of ANODEs.
result SONODEs and ANODEs can learn higher order dynamics efficiently.
In this paper we continue to study equivariant pencil liftings and differential operators on the algebra of densities. We emphasize the role that the geometry of the extended manifold plays. Firstly we consider basic examples. We give a projective line of diff()-equivariant pencil liftings for first order operators,…
Paper analyzes double twist knots using adjoint hyperbolic torsion polynomial.
problem Determining the genus and fibering of double twist knots.
method Uses adjoint hyperbolic torsion polynomial to analyze double twist knots.
result The adjoint hyperbolic torsion polynomial determines the genus and fibering of double twist knots.
Derives adjoint formulas for matrix operations and applies them to specific cases.
problem Computing adjoints for matrix operations and specific matrix types.
method Derives adjoint formulas for matrix operations and applies them to specific cases.
result Closed-form expressions for adjoints in specific matrix types.
A new framework models uncertainty in structured temporal data using SDEs and neural networks.
problem Uncertainty quantification in machine learning applications involving structured and temporal data.
method Integrates stochastic differential equations (SDEs) with deep generative models in a variational autoencoder framework.
result Improves uncertainty quantification in machine learning applications involving structured and temporal data.
Study of filtering and smoothing in submanifolds of Euclidean space.
problem Filtering and smoothing in continuous-discrete time on submanifolds.
method Formal expressions and projection approach for prediction and smoothing.
result Agreement with classical results for prediction, differences for smoothing.
This article is a continuation of my former article "On Connectivity Spaces". After some brief historical references relating to the subject, separation spaces and then adjoint notions of connective representation and connective foliation are developed. The connectivity order previously defined only in the finite case …
Unified approach to Merton's portfolio problem using Pontryagin's principles.
problem Optimizing consumption and investment strategies in financial portfolios.
method PG-DPO framework combining neural networks with Pontryagin's maximum principle.
result Locally optimal policies closely tied to classical stochastic control.
Explicit formula for Reidemeister torsion of two-bridge knots.
problem Calculating Reidemeister torsion for two-bridge knots.
method Provided an explicit formula and proved vanishing identities.
result Adjoint Reidemeister torsion satisfies vanishing identities.
By now it is well established that the quantum dimensions of descendants of the adjoint representation can be described in a universal form, independent of a particular family of simple Lie algebras. The Rosso-Jones formula then implies a universal description of the adjoint knot polynomials for torus knots, which in p…
The paper explores self-adjointness of Laplace-Beltrami operator on special geometric manifolds.
problem Characterizing self-adjoint extensions of the Laplace-Beltrami operator on -Grushin manifolds.
method Introducing an exotic calculus of pseudodifferential operators adapted to the geometry of the singularity.
result Criterion for essential self-adjointness and determination of several self-adjoint extensions.
Study extends Vogel's universality to torus knots in adjoint representation.
problem Applying Vogel's universality to knot invariants in adjoint representation theory.
method Extending Vogel's parameters to include torus knots and focusing on with odd .
result Unified description of adjoint invariants for torus knots with odd .
Study Hamiltonian diffeomorphisms on symplectic manifolds and properties of invariant convex functions.
problem Properties of invariant convex functions under Hamiltonian diffeomorphisms.
method Analysis of the adjoint action and properties of invariant convex functions.
result Continuous convex functions invariant under Hamiltonian diffeomorphisms are also invariant under strict rearrangements.
Extends adjoint representation concept to higher Lie groupoids.
problem Defining adjoint representation for higher Lie groupoids.
method Generalizes standard construction to higher Lie groupoids using simplicial vector bundles.
result Adjoint representation up to homotopy is well-defined and unique.
New knot theory module shows torsion-ness in number theory.
problem Torsion-ness of Selmer modules in Galois representations.
method Introducing adjoint homological Selmer module for SL-representations of knot groups.
result Finitely generated torsion-ness of the new Selmer module.