As is well-known, the Witten deformation of the De Rham complex computes the De Rham cohomology. In this paper we study the Witten deformation on a noncompact manifold and restrict it to differential forms which behave polynomially near infinity. Such polynomial differential forms naturally appear on manifolds with a c…
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48 results for “polynomial differential forms”
On characteristic forms of positive vector bundles, mixed discriminants and pushforward identitiesmath.AG
The paper proves positivity of characteristic forms for certain vector bundles.
problem Characterizing positivity conditions for vector bundles.
method Operator theory, pushforward identities, and differential forms.
result Schur polynomials in Chern forms of Nakano and Griffiths positive vector bundles are positive as differential forms.
We simplify complex geometric structures for contracting measurable systems.
problem Normal forms for contracting measurable cocycles and foliations.
method Differential-geometric approach to obtain resonance polynomial normal forms via changes of coordinates.
result Obtained nonstationary invariant differential-geometric structures for contracting systems and foliations.
New link homologies categorify Jones polynomial at odd prime powers.
problem Categorify Jones polynomial at odd prime powers.
method Specialize Cautis differential to positive integers.
result Non-isomorphic link homologies for odd primes.
Constructs finite element spaces for -forms, excluding one subspace.
problem Constructing finite element spaces for -forms.
method Piecewise polynomial finite element spaces for all natural subspaces of -forms, excluding one subspace.
result Recovers known finite element spaces and introduces new ones.
New proof of duality in finite element exterior calculus.
problem Proving duality relationship between finite element spaces.
method Alternate construction of finite element spaces and new basis-free proof using modified Hodge star operator.
result A new, basis-free proof of the duality relationship.
The boundary value problem for Laplacian on differential forms and conformally Einstein infinitymath.DG
We completely resolve the boundary value problem for differential forms and conformally Einstein infinity in terms of the dual Hahn polynomials. Consequently, we produce explicit formulas for the Branson-Gover operators on Einstein manifolds and prove their representation as a product of second order operators. This le…
This paper is devoted to the characterization of differentially flat nonlinear systems in implicit representation, after elimination of the input variables, in the differential geometric framework of manifolds of jets of infinite order. We extend the notion of Lie-Bäcklund equivalence, introduced in Fliess et al. (1999…
Study Z2 harmonic functions with singularities on flat space.
problem Understanding Z2 harmonic functions with point singularities.
method Analyzes Z2 harmonic functions on with point singularities.
result Characterizes Z2 harmonic functions on with point singularities.
Simpler equations derived for knot polynomials coefficients, forming a ring.
problem Complexity of knot polynomials colored with symmetric representations.
method Deriving two difference equations for quantum C-polynomials coefficients.
result Quantum C-polynomials form a ring and are much simpler than colored polynomials.
Study conformal symmetry breaking operators on differential forms.
problem Mapping differential forms on to .
method Apply F-method to derive formulas for intertwining operators.
result Explicit formulas for symmetry breaking differential operators.
Conformal operators on weighted forms; their decomposition and null space on Einstein manifoldsmath.DG
There is a class of Laplacian like conformally invariant differential operators on differential forms which may be considered the generalisation to differential forms of the conformally invariant powers of the Laplacian known as the Paneitz and GJMS operators. On conformally Einstein manifolds we give explic…
Classifies conformally covariant operators between differential forms on spheres.
problem Classifying conformally covariant differential operators between differential forms on spheres.
method F-method based on algebraic Fourier transform of Verma modules, extended to matrix-valued case.
result Explicit formulæ for conformally covariant differential operators in flat coordinates.
The note answers a question about Betti numbers for 1D Euclidean space.
problem Understanding Betti numbers for vector fields and differential forms in 1D Euclidean space.
method Using Euler vector field and Lie superalgebra structure.
result The Betti numbers are 1 for the case where primary and secondary weights are equal.
Develops unisolvent weights for Nédélec second family finite elements in 2D.
problem Finding efficient degrees of freedom for Nédélec second family finite elements.
method Uses techniques of homological algebra to obtain degrees of freedom for differential forms.
result Provides a family of unisolvent and minimal physical degrees of freedom for Nédélec second family finite elements.
Proves a formula for push-forward of polynomial Chern forms in universal vector bundles.
problem Positivity of characteristic forms in vector bundles.
method Explicit computation of Chern curvature and use of flag bundles.
result Positivity of polynomials in Chern forms for Griffiths semipositive bundles.
Numerical experiments support conjecture about opers and nonabelian Hodge.
problem Testing predictions of Gaiotto-Moore-Neitzke and Gaiotto conjectures.
method Numerical experiments on polynomial holomorphic differentials.
result Supports conjectural formulas for Stokes data and Hitchin metric tensor.
Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear termsmath.NA
New method learns low-dimensional models for systems with non-polynomial terms.
problem Modeling systems with non-polynomial nonlinear terms that are spatially local and given in analytic form.
method Non-intrusive model reduction method that learns operators for linear and polynomially nonlinear dynamics via a least-squares problem incorporating given non-polynomial terms.
result Comparable accuracy to intrusive methods that require full knowledge of governing equations.
Develops VPINNs for solving PDEs with reduced training cost and improved accuracy.
problem Solving partial differential equations efficiently and accurately.
method Integrates variational forms of PDEs into neural network loss functions, using Legendre polynomials as test spaces.
result VPINNs outperform PINNs in terms of accuracy and speed for solving PDEs.
Flag area measuresmath.DG
New flag area measures constructed via integration over differential forms.
problem Constructing new flag area measures in high-dimensional spaces.
method Using local parallel sets and integration over the normal cycle of differential forms.
result Flag area measures span the space of all smooth SO(n)-covariant flag area measures.
Criteria for smoothness of ambiskew polynomial rings.
problem Smoothness of ambiskew polynomial rings.
method Determined sufficient criteria for differential smoothness.
result Criteria for differential smoothness of ambiskew polynomial rings.
The Hessian Topology is a subject with interesting relations with some classical problems of analysis and geometry. In this article we prove a conjecture on this subject stated by V.I. Arnold concerning the number of connected components of hyperbolic homogeneous polynomials of degree . The proof is constructive and…
New sampling method for heavy-tailed distributions using Langevin Algorithm.
problem Sampling from heavy-tailed distributions efficiently.
method Transformed Unadjusted Langevin Algorithm on specific transformations.
result Polynomial-order oracle complexities for certain heavy-tailed densities.
Let M be a manifold and g a Lie algebra acting on M. Differential forms Omega(M) carry a natural action of Lie derivatives L(x) and contractions I(x) of fundamental vector fields for x \in g. Contractions (anti-) commute with each other, [I(x), I(y)]=0. Together with the de Rham differential, they satisfy the Cartan's …
We prove regularity results up to the boundary for time independent generalized Maxwell equations on Riemannian manifolds with boundary using the calculus of alternating differential forms. We discuss homogeneous and inhomogeneous boundary data and show 'polynomially weighted' regularity in exterior domains as well.
On the differential smoothness of 3-dimensional skew polynomial algebras and diffusion algebrasmath.DG
The paper examines differential smoothness in specific algebra types.
problem Differential smoothness in 3D skew polynomial algebras and diffusion algebras.
method Analyzes the properties of 3D skew polynomial algebras and diffusion algebras.
result Provides insights into the differential smoothness of these algebra types.
The paper examines differential smoothness in skew PBW extensions over polynomial rings.
problem Differential smoothness in skew PBW extensions over polynomial rings.
method Investigation of skew PBW extensions over commutative polynomial rings.
result Results on differential smoothness for skew PBW extensions over polynomial rings.
New finite element method for complex forms in any dimension.
problem Discretization of complex forms in arbitrary dimensions.
method Finite element discretization of -form-valued -forms on triangulations.
result Generalizes existing finite element methods for various tensor fields.
Extends gradient-based optimization to spline functions.
problem Limitations of standard differentiable programming methods.
method Derives Jacobian of spline functions and uses it in predictive models.
result Improved performance in various applications.
On the Polynomial and Exponential Decay of Eigen-Forms of Generalized Time-Harmonic Maxwell Problemsmath.AP
We prove polynomial and exponential decay at infinity of eigen-vectors of partial differential operators related to radiation problems for time-harmonic generalized Maxwell systems in an exterior domain with non-smooth inhomogeneous, anisotropic coefficients converging near infinity with a certain rate towards the iden…
Let G be a finite group of complex n by n unitary matrices generated by reflections acting on C^n. Let R be the ring of invariant polynomials, and χbe a multiplicative character of G. Let Ω^χbe the R-module of χ-invariant differential forms. We define a multiplication in Ω^χand show that under this multiplication Ω^χha…
Investigates differential smoothness of 3D skew polynomial rings.
problem Differential smoothness of 3D skew polynomial rings.
method Analyzes Bell and Smith's characterization of 3D skew polynomial rings.
result Provides insights into the differential smoothness of these rings.
Algorithm constructs integrals of hyperexponential 1-forms.
problem Computing integrals of hyperexponential 1-forms.
method Using Schanuel's conjecture, decomposes into .
result Algorithm constructs a basis of cohomology of differential 1-forms.
The study describes Nijenhuis operators with specific properties.
problem Characterizing Nijenhuis operators with functional independence and determinant constraints.
method Proving the general form and describing the specific case of Nijenhuis operators.
result Complete description of Nijenhuis operators with nondegenerate determinant.
Paper proves positivity of Chern-Weil forms for certain vector bundles.
problem Proving positivity of Chern-Weil forms for Griffiths semipositive vector bundles.
method Analyzing characteristic differential forms and Schur polynomials.
result Positivity of established.
We recall the main facts about the odd Laplacian acting on half-densities on an odd symplectic manifold and discuss a homological interpretation for it suggested recently by P. {Š}evera. We study the relationship of odd symplectic geometry with classical objects. We show that the Berezinian of a canonical transformatio…
New proof of Alesker's Irreducibility Theorem using localization techniques.
problem Representing polynomial valuations on convex bodies.
method Introducing a localization technique for polynomial valuations and reducing to a representation problem for differential forms.
result Smooth and translation invariant valuations are representable by integration with the normal cycle.
Defines Reidemeister torsion form on 3-manifold character varieties.
problem Character variety singularities and vanishing torsion.
method Differential form on character variety, Alexander polynomial, Culler-Shalen theory.
result Torsion vanishes only at singular points, related to reducibility and Euler characteristic.
Paper calculates Racah matrices for 3-strand knots, validating conjectures.
problem Systematic description of colored knot polynomials.
method Highest weight method with Gelfand-Tseitlin tables.
result Explicit Racah matrices and polynomials for 3-strand knots up to 10 crossings.
Maps complex plane polynomials to light-like polygons in Einstein Universe.
problem Mapping between complex plane polynomials and light-like polygons.
method Constructs geometric homeomorphism between moduli spaces.
result Found minimal Lagrangian maps between ideal polygons.
Grassmannians Gr(N-1,N+1), closed differential N-1 forms and N-dimensional integrable systemsnlin.SI
Integrable flows on the Grassmannians Gr(N-1,N+1) are defined by the requirement of closedness of the differential N-1 forms of rank N-1 naturally associated with Gr(N-1,N+1). Gauge-invariant parts of these flows, given by the systems of the N-1 quasi-linear differential equations, describe coisotropic deform…
The current article studies certain problems related to complex cycles of holomorphic foliations with singularities in the complex plane. We focus on the case when polynomial differential one-form gives rise to a foliation by Riemann surfaces. In this setting, a complex cycle is defined as a nontrivial element of the f…
We prove that Chern-Weil forms are the only natural differential forms associated to a connection on a principal G-bundle. We use the homotopy theory of simplicial sheaves on smooth manifolds to formulate the theorem and set up the proof. Other arguments come from classical invariant theory. We identify the Weil algebr…
KNTZ trick simplifies knot polynomial calculations for twist knots.
problem Completing the structure of differential expansion for twist knots.
method Converting arborescent evolution matrix into triangular form.
result Conjecture for triangular matrix in non-rectangular case.
Categorifies colored Jones polynomial at roots of unity.
problem Categorification of colored Jones polynomial.
method Differential on triply-graded homology, compatible with p-differential structure.
result Categorification of the colored Jones polynomial at a root of unity.
Computes differential invariants for conformal metrics.
problem Local recognition of conformal structures.
method Computes Hilbert polynomial and Poincare function.
result Resolves the local recognition problem for conformal structures.
New duality relationships between finite element spaces on spheres derived from Hodge theory.
problem Developing finite element methods for differential forms on spheres.
method Showed that the duality relationships between finite element spaces are Hodge duality on the sphere, providing explicit correspondences.
result Explicit correspondences between finite element spaces and differential forms on the sphere, including new pointwise duality isomorphisms.
Study on unique vortex equations and their geometric implications.
problem Uniqueness of vortex equations involving entire functions.
method Analyzing entire functions on the complex plane and showing geometric applications.
result Uniqueness of harmonic maps and affine spherical immersions with polynomial differential constraints, but failure for non-polynomial entire functions.