A new method for computing shape gradients in FSI problems with non-matching meshes.
problem Computing shape gradients in fluid-structure interaction problems with non-matching meshes.
method Partitioned solution procedure using black-box adjoint solvers, augmented target functions, and coupling fields.
result Accurate shape gradients computed with reduced formulations for computational efficiency.
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.
Explicit adjoint group description for Coxeter quandles.
problem Understanding the adjoint group structure of Coxeter quandles.
method Explicit descriptions and constructions of adjoint groups, using central extensions and 2-cocycles.
result The adjoint group of a Coxeter quandle is an intermediate group between the Coxeter group and its Artin group, with specific properties related to commutator subgroups and root systems.
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 sl(2,R). 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.
New degree theory for perturbed self-adjoint operators.
problem Equivariant gradient perturbations of unbounded self-adjoint operators.
method Equivariant gradient degree for perturbations of self-adjoint operators with discrete spectrum.
result Definition and application of new degree theory.
Prove integrality of genus-g indices with adjoint Reidemeister torsions for twist knots and meridians.
problem Prove integrality of genus-g 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-g indices with adjoint Reidemeister torsions for twist knots and meridians. 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.
Essential self-adjointness proven for sub-Laplacians on sub-Riemannian manifolds.
problem Proving essential self-adjointness for sub-Laplacians on sub-Riemannian manifolds.
method General essential self-adjointness criterion for sub-Laplacians on complete sub-Riemannian manifolds, defined with respect to singular measures.
result Intrinsic sub-Laplacian is essentially self-adjoint on equiregular connected components of a sub-Riemannian manifold under mild regularity assumptions.
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.
Explains complex adjoint orbits in Lie theory and geometry.
problem Understanding adjoint orbits in Lie theory and geometry.
method Expository introduction to adjoint orbits of complex semisimple groups.
result Provides insights into properties of semisimple and nilpotent orbits.
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.
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 α-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.
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.
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 T[m,n] and focusing on T[4,n] with odd n. result Unified description of adjoint invariants for torus knots T[4,n] with odd n. 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 SL2-representations of knot groups. result Finitely generated torsion-ness of the new Selmer module.
Study shows essential self-adjointness of wave operators in Lorentzian settings.
problem Essential self-adjointness of wave operators in Lorentzian scattering spaces.
method Using a Fredholm framework to invert the spectral family and refine conclusions.
result Dense range in L^2 for the wave operator acting on an appropriate subdomain.
We consider magnetic geodesic flows of the normal metrics on a class of homogeneous spaces, in particular (co)adjoint orbits of compact Lie groups. We give the proof of the non-commutative integrability of flows and show, in addition, for the case of (co)adjoint orbits, the usual Liouville integrability by means of ana…
The study confirms essential self-adjointness for certain differential operators on manifolds.
problem Essential self-adjointness of differential operators on closed manifolds.
method Analyzing the Hamiltonian flow of the symbol of differential operators.
result The conjecture that certain differential operators are essentially self-adjoint if their Hamiltonian flow is complete.
Paper proves index theorem for self-adjoint elliptic boundary problems.
problem Proving index theorem for self-adjoint elliptic boundary problems.
method Topological and pseudo-differential methods, generalized Atiyah-Singer approach.
result Removed technical assumption to prove index theorem.
Adjoint sampler targets infinite-dimensional function spaces for efficient sampling.
problem Limited theory and algorithms for sampling infinite-dimensional function spaces.
method Adjoint Sampler for infinite-dimensional function spaces based on stochastic maximum principle.
result FAS achieves superior performance in synthetic and real systems.
Geometrically connects adjoint and coadjoint orbits of semidirect products.
problem Establishing a bijection between adjoint and coadjoint orbits for semidirect products.
method Proving a geometric bijection using specific subgroup conditions and homotopy types.
result Homotopy types of orbits in bijection are the same under certain conditions.
Let Δ be a linear differential operator acting on the space of densities of a given weight $\lo$ on a manifold M. One can consider a pencil of operators $\hPi(Δ)=\{Δ_ł\}$ passing through the operator Δ such that any Δł is a linear differential operator acting on densities of weight ł. This pencil can be iden…
We study the Gaffney Laplacian on a vector bundle equipped with a compatible metric and connection over a Riemannian manifold that is possibly geodesically incomplete. Under the hypothesis that the Cauchy boundary is polar, we demonstrate the self-adjointness of this Laplacian. Furthermore, we show that negligible boun…
I show that the adjoint variety of the complex special linear group is rigid to order three.
The study finds Lagrangian submanifolds in adjoint semisimple orbits for real forms.
problem Characterizing Lagrangian submanifolds in adjoint semisimple orbits.
method Analyzing real flags and orbits of real forms with respect to symplectic forms.
result Classification of infinitesimally tight Lagrangian submanifolds in the compact case and Lagrangian submanifolds in the complex case.
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 give explicit formulas for the adjoint twisted Alexander polynomial and the nonabelian Reidemeister torsion of genus one two-bridge knots.
Let G be a real compact connected simple Lie group, and g its Lie algebra. We study the problem of determining, from root data, when a sum of adjoint orbits in g, or a product of conjugacy classes in G, contains an open set. Our general methods allow us to determine exactly which sums of adjoint orbits in su(m) and pro…
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.
Study geodesics on SL(n,R) adjoint orbits.
problem Geodesics on adjoint orbits of SL(n,R). method Translate problem into tangent bundle of SO(n)-flag manifolds, use Sasaki metric and Lie Theory. result Explicit description of geodesics families in SL(2,R). Study modular class of Lie ∞-algebroids and their adjoint actions.
problem Understanding the modular class and adjoint actions of Lie ∞-algebroids.
method Equivalence of descriptions, homotopy invariance, explicit actions and dualities.
result Homotopy invariance of modular classes and explicit adjoint actions.
We study the hyperkaehler geometry of a regular semisimple adjoint orbit of SL(k,C) via the algebraic geometry of the corresponding reducible spectral curve.
It is known that the Schrödinger flow on a complex Grassmann manifold is equivalent to the matrix non-linear Schrödinger equation and the Ferapontov flow on a principal Adjoint U(n)-orbit is equivalent to the n-wave equation. In this paper, we give a systematic method to construct integrable geometric curve flows on …
In this note we define the stabilizer group of any adjoint-invariant l-form on a complex simple Lie algebra. This result partially extend a previous result by Kable.
Study of Green function and Laplacians on polyhedral surfaces, focusing on genus two with a conical point.
problem Analyzing the behavior of Green function and self-adjoint Laplacians on polyhedral surfaces.
method Explicit construction of a basis in the kernel of the adjoint Laplacian, computation of S-matrix, study of various self-adjoint extensions.
result The behavior of the S-matrix at the zero value of the spectral parameter is sensitive to the geometry of the polyhedron.
Essential self-adjointness proved for perturbed quadharmonic operators on Riemannian manifolds.
problem Proving essential self-adjointness for perturbed quadharmonic operators.
method Using bounded geometry assumptions and a non-positive potential function.
result Essential self-adjointness condition established for perturbed quadharmonic operators.
We consider first-order differential operators with locally bounded measurable coefficients on vector bundles with measurable coefficient metrics. Under a mild set of assumptions, we demonstrate the equivalence between the essential self-adjointness of such operators to a negligible boundary property. When the operator…
Adjoint SA speeds up bioprocess parameter learning.
problem Challenges in digital twin development for biomanufacturing.
method Adjoint sensitivity analysis on multi-scale enzymatic reaction networks.
result Resilient sensitivities reveal bioprocess regulatory mechanisms.
Given a compact Lie group, endowed with a bi-invariant Riemannian metric, its complexification inherits a Kaehler structure having twice the kinetic energy of the metric as its potential, and Kaehler reduction with reference to the adjoint action yields a stratified Kaehler structure on the resulting adjoint quotient. …
Proves a conjecture about factorization homology leading to the cobordism hypothesis.
problem Proving a conjecture about factorization homology.
method Assuming a conjecture about factorization homology with adjoints.
result Proves the cobordism hypothesis.
The paper proves positivity preservation and self-adjointness for Schrödinger operators on incomplete Riemannian manifolds.
problem Positivity preservation and self-adjointness for Schrödinger-type operators on incomplete Riemannian manifolds.
method Control of potential behavior near the Cauchy boundary, essential self-adjointness proof, core of smooth compactly supported functions.
result Positivity preservation and essential self-adjointness of Schrödinger operators on Lp functions on incomplete Riemannian manifolds. ASBS improves sampling from Boltzmann distributions without importance weighting.
problem Sampling from Boltzmann distributions with known energies but unknown samples.
method Adjoint Schrödinger Bridge Sampler using kinetic-optimal transportation.
result ASBS achieves scalable and efficient sampling without importance weighting.