Higher order Lagrange-Poincar, and Hamilton-Poincar, reductions

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In this paper, we present metriplectic extensions of the Lie–Poisson dynamics of fluid flows over the unit annulus for both the Lebesgue [Formula: see text]-pairing and the Sobolev [Formula: see text]-pairing. Several examples are provided including 2D Nav ...

In this thesis, we propose model order reduction techniques for high-dimensional PDEs that preserve structures of the original problems and develop a closure modeling framework leveraging the Mori-Zwanzig formalism and recurrent neural networks. Since high ...

Modern manufacturing engineering is based on a ``design-through-analysis'' workflow. According to this paradigm, a prototype is first designed with Computer-aided-design (CAD) software and then finalized by simulating its physical behavior, which usually i ...

We develop structure-preserving reduced basis methods for a large class of nondissipative problems by resorting to their formulation as Hamiltonian dynamical systems. With this perspective, the phase space is naturally endowed with a Poisson manifold struc ...

We illustrate the potential of a very high energy lepton collider (from 10 to 30 TeV center of mass energy) to explore new physics indirectly in the vector boson fusion double Higgs production process and in direct diboson production at high energy. Double ...

In this paper we propose a dynamical low-rank strategy for the approximation of second order wave equations with random parameters. The governing equation is rewritten in Hamiltonian form and the approximate solution is expanded over a set of 2S dynamical ...

Ratios of isospin amplitudes in hadron decays are a useful probe of the interplay between weak and strong interactions and allow searches for physics beyond the standard model. We present the first results on isospin amplitudes in b-baryon decays, using da ...

We give a generalization of toric symplectic geometry to Poisson manifolds which are symplectic away from a collection of hypersurfaces forming a normal crossing configuration. We introduce the tropical momentum map, which takes values in a generalization ...

In this work, we focus on the Dynamical Low Rank (DLR) approximation of PDEs equations with random parameters. This can be interpreted as a reduced basis method, where the approximate solution is expanded in separable form over a set of few deterministic b ...

A classical theorem of Frankel for compact Kahler manifolds states that a Kahler S-1-action is Hamiltonian if and only if it has fixed points. We prove a metatheorem which says that when the Hodge theory holds on non-compact manifolds, Frankel's theorem st ...

Let Q denote a smooth manifold acted upon smoothly by a Lie group G. The G-action lifts to an action on the total space TQ of the cotangent bundle of Q and hence on the standard symplectic Poisson algebra of smooth functions on TQ. The Poisson algebra of ...

A theorem of Drinfel'd (Drinfel'd (1993)) classifies the Poisson homogeneous spaces of a Poisson Lie group (G,πG) via a special class of Lagrangian subalgebras of the Drinfel'd double of its Lie bialgebra. This result is extended in Liu et al. (1998) to a ...

The notions of Poisson Lie group and Poisson homogeneous space are extended to the Dirac category. The theorem of Drinfeld on the one-to-one correspondence between Poisson homogeneous spaces of a Poisson Lie group and a special class of Lagrangian subalgeb ...

The equations of motion are derived for the dynamical folding of charged molecular strands (such as DNA) modeled as flexible continuous filamentary distributions of interacting rigid charge conformations. The new feature is that these equations are nonloca ...

We formulate Euler-Poincare and Lagrange-Poincare equations for systems with broken symmetry. We specialize the general theory to present explicit equations of motion for nematic systems, ranging from single nematic molecules to biaxial liquid crystals. Th ...

Quantum dots (QDs) of high symmetry (e.g., C3v) have degenerate bright exciton states, unlike QDs of C2v symmetry, making them intrinsically suitable for the generation of entangled photon pairs. Deviations from C3v symmetry are detected in real QDs by pol ...

Every action on a Poisson manifold by Poisson diffeomorphisms lifts to a Hamiltonian action on its symplectic groupoid which has a canonically defined momentum map. We study various properties of this momentum map as well as its use in reduction. ...

This paper develops the theory of affine Euler-Poincare and affine Lie-Poisson reductions and applies these processes to various examples of complex fluids, including Yang-Mills and Hall magnetohydrodynamics for fluids and superfluids, spin glasses, microf ...

For a given skew symmetric real n x n matrix N, the bracket X, Y = XNY - YNX defines a Lie algebra structure on the space Sym(n, N) of symmetric n x n real matrices and hence a corresponding Lie-Poisson structure. The purpose of this paper is to inves ...

Higher order Lagrange-Poincar, and Hamilton-Poincar, reductions

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Area preserving flows and metriplectic dynamics on an annulus

Structure-preserving approaches and data-driven closure modeling for model order reduction

Stabilized isogeometric discretizations on trimmed and union geometries, and weak imposition of the boundary conditions for the Darcy flow

Structure-Preserving Reduced Basis Methods For Poisson Systems

Two paths towards precision at a very high energy lepton collider

Symplectic dynamical low rank approximation of wave equations with random parameters

Isospin Amplitudes in Λ0b→J/ψΛ(Σ0) and Ξ0b→J/ψΞ0(Λ) Decays

Structure-Preserving Reduced Basis Methods for Hamiltonian Systems with a State-dependent Poisson Structure

The tropical momentum map: a classification of toric log symplectic manifolds

Dynamical Low Rank approximation of PDEs with random parameters

L-2-cohomology and complete Hamiltonian manifolds

Extensions of Lie-Rinehart algebras and cotangent bundle reduction

Dirac Group(oid)s and Their Homogeneous Spaces

Dirac Lie Groups, Dirac Homogeneous Spaces and the Theorem of Drinfeld

Symmetry Reduced Dynamics of Charged Molecular Strands

Reduction theory for symmetry breaking with applications to nematic systems

Fine structure of exciton complexes in high-symmetry quantum dots: Effects of symmetry breaking and symmetry elevation

The Momentum Map In Poisson Geometry

The geometric structure of complex fluids

A Class of Integrable Flows on the Space of Symmetric Matrices