Lagrangian formalism on graded manifolds is phrased in terms of the Grassmann-graded variational bicomplex, generalizing the familiar variational bicomplex for even Lagrangian systems on fiber bundles.
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48 results for “Variational bicomplex”
We present a family of complexes playing the same role, for homogeneous variational problems, that the horizontal parts of the variational bicomplex play for variational problems on a fibred manifold. We show that, modulo certain pullbacks, each of these complexes (apart from the first one) is globally exact. All the c…
We show how the homogeneous variational bicomplex provides a useful formalism for describing a number of properties of single-integral variational problems, and we introduce a subsequence of one of the rows of the bicomplex which is locally exact with respect to the variational derivative. We are therefore able to reco…
We obtain the cohomology of the variational bicomplex on the infinite order jet space of a smooth fiber bundle in the class of exterior forms of finite jet order. This provides a solution of the global inverse problem of the calculus of variations of finite order on fiber bundles.
Defines a filtration on variational bicomplex for concise functional form conditions.
problem Expressing functional form vanishing conditions concisely.
method Introduces a filtration on the variational bicomplex and studies its properties.
result Graded components of the filtration inherit module structures, simplifying functional form conditions.
We study the obstruction to the exactness of the variational complex for a field theory on an affine bundle.
Constructs Lepage equivalents for arbitrary-order Lagrangians.
problem Creating Lepage equivalents for complex Lagrangians.
method Uses variational bicomplex and symmetric linear connections to construct Lepage equivalents satisfying the closure property.
result Shows how to extend global Lepage equivalents to ones satisfying the closure property.
In this work we show that the systems of balance equations (balance systems) of continuum thermodynamics occupy a natural place in the variational bicomplex formalism. We apply the vertical homotopy decomposition to get a local splitting (in a convenient domain) of a general balance system as the sum of a Lagrangian pa…
We use functions of a bicomplex variable to unify the existing constructions of harmonic morphisms from a 3-dimensional Euclidean or pseudo-Euclidean space to a Riemannian or Lorentzian surface. This is done by using the notion of complex-harmonic morphism between complex-Riemannian manifolds and showing how these are …
Lagrangian contact supersymmetries (depending on derivatives of arbitrary order) are treated in very general setting. The cohomology of the variational bicomplex on an arbitrary graded manifold and the iterated cohomology of a generic nilpotent contact supersymmetry are computed. In particular, the first variational fo…
New variational principle found for non-variational differential equations.
problem Non-variational differential equations without variational multipliers.
method Connecting functional forms with antiexact differential forms to identify obstructions.
result Formulation of variational problem for non-variational equations.
A colored Khovanov bicomplexmath.GT
Proves existence of colored Khovanov bicomplex linking Jones polynomial.
problem Establishing a bicomplex for colored Jones polynomials.
method Develops a tri-graded bicomplex with specific gradings.
result The graded Euler characteristic of the total complex equals the colored Jones polynomial.
In this paper, we denone the generalized bicomplex numbers and give some algebraic properties of them. Also, we show that some hyperquadrics in R4 and R42 are Lie groups by using generalized bicomplex number product and obtain Lie algebras of these Lie groups. Morever, by using tensor product surfaces, we determine som…
The inverse problem of the calculus of variations asks whether a given system of partial differential equations (PDEs) admits a variational formulation. We show that the existence of a presymplectic form in the variational bicomplex, when horizontally closed on solutions, allows us to construct a variational formulatio…
This paper provides a description of an algebraic setting for the Lagrangian formalism over graded algebras and is intended as the necessary first step towards the noncommutative C-spectral sequence (variational bicomplex). A noncommutative version of integration procedure, the notion of adjoint operator, Green's formu…
Homotopy momentum map extends Noether's theorem in general relativity.
problem Extending Noether's theorem to spacetime vector fields.
method Using homotopy momentum map and -algebras.
result Extension of conserved currents to spacetime vector fields.
We construct a bicomplex for the categorification of the colored Jones polynomial. This work is motivated by the problem suggested by Anna Beliakova and Stephan Wehrli who discussed the categorification of the colored Jones polynomial in their paper.
Bi-flat F-structures link to differential bicomplexes and Gauss-Manin connections.
problem Understanding the geometric structure of bi-flat F-structures.
method Showed bi-flat F-structures define a differential bicomplex and relate to Gauss-Manin connections.
result Flat connections associated with bi-flat structures can be identified with Levi-Civita connections of flat metrics.
Informed by recent work on tensor singular value decomposition and circulant algebra matrices, this paper presents a new theoretical bridge that unifies the hypercomplex and tensor-based approaches to singular value decomposition and robust principal component analysis. We begin our work by extending the principal comp…
We introduce colored Jones polynomials of nanowords and their categorification. We also prove the existence of a Khovanov-type bicomplex which has three grades.
Defines new bi-flat structures from integrable systems and flat coordinates.
problem Creating new bi-flat structures from integrable systems.
method Combining Frölicher-Nijenhuis bicomplex with Lauricella bi-flat structures.
result Defines multi-parameter families of Lauricella bi-flat structures.
We derive simple explicit formulas for the character of a cycle in the Connes' (b,B)-bicomplex of cyclic cohomology and give applications to the Fredholm modules and equivariant characteristic classes.
A generic degenerate Lagrangian system of even and odd variables on an arbitrary smooth manifold is examined in terms of the Grassmann-graded variational bicomplex. Its Euler-Lagrange operator obeys Noether identities which need not be independent, but satisfy first-stage Noether identities, and so on. However, non-tri…
Differential forms on an odd symplectic manifold form a bicomplex: one differential is the wedge product with the symplectic form and the other is de Rham differential. In the corresponding spectral sequence the next differential turns out to be the Batalin-Vilkoviski operator.
In this note we define a notion of Courant pair as a Courant algebra over the Lie algebra of linear derivations on an associative algebra. We study formal deformations of Courant pairs by constructing a cohomology bicomplex with coefficients in a module from the cochain complexes defining Hochschild cohomology and Leib…
We introduce a bicomplex which computes the triple cohomology of Lie--Rinehart algebras. We prove that the triple cohomology is isomorphic to the Rinehart cohomology \cite{Ri} provided the Lie--Rinehart algebra is projective over the corresponding commutative algebra. As an application we construct a canonical class in…
In this paper we study the local description of spaces of forms on transitive Lie algebroids. We use this local description to introduce global structures like metrics, -Hodge operation and integration along the algebraic part of the transitive Lie algebroid (its kernel). We construct a Čech-de Rham bicomplex wit…
Spectral sequences of colored Jones polynomials, colored Rasmussen invariants and nanophrasesmath.GT
We introduce three spectral sequences which give some expressions of colored Jones polynomials. Each spectral sequence contains a Khovanov-type homology groups. Two of them are derived from a bicomplex of the colored Jones polynomial. The other is the spectral sequence that deduces a colored Rasmussen invariant of link…
The paper focuses on various properties and applications of the homotopy operator, which occurs in the Poincaré lemma. In the first part, an abstract operator calculus is constructed, where the exterior derivative is an abstract derivative and the homotopy operator plays the role of an abstract integral. This operator …
Develops homotopies for Lagrangian field theory using advanced algebraic structures.
problem Formulating a consistent framework for Lagrangian field theory.
method Introduces algebras and homotopies to enrich the Batalin-Vilkovisky framework.
result Provides an explicit lift of the Batalin-Vilkovisky framework to local forms.
Simplified Khovanov-Rozansky operators for link invariants.
problem Complexity in Khovanov-Rozansky knot invariants.
method Substituted matrix factorization with simple operators and conjugations.
result Global reductions for simplifying link invariants.
Simplified KR polynomial for bipartite links reduces to tensor products of vector spaces.
problem Complexity reduction of Khovanov-Rozansky polynomial for bipartite links.
method Local reduction of matrix factorizations to planar cycles and simplification to vector spaces.
result KR polynomial for bipartite links simplifies to tensor products of vector spaces.
Model structures on multicomplexes help study complex geometry.
problem Understanding homotopy types of complex manifolds.
method Model category structures on -multicomplexes with weak equivalences induced by quasi-isomorphisms.
result Establishes a basis for studying almost and generalized complex manifolds.
Holographic coordinatesmath.DG
The Laplace equation in the two-dimensional Euclidean plane is considered in the context of the inverse stereographic projection. The Lie algebra of the conformal group as the symmetry group of the Laplace equation can be represented solely in terms of the solutions and derivatives of the solutions of the Laplace equat…
Develops Poisson structures on weak Sobolev loop spaces for integrable systems.
problem Analyzing integrable systems on low regularity loop spaces.
method Extending Mokhov's constructions to weak Sobolev spaces, constructing presymplectic and Poisson structures.
result Valid Poisson structures and deformations for weak Sobolev loops, extending Hamiltonian formalisms.
We associate to each infinite primitive Lie pseudogroup a Hopf algebra of `transverse symmetries', by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed as a `quantum group' counterpart of the infinite-dimensional primitive Lie algeb…
Study quantum aspects of 1-form symmetries using BV-BRST cohomology.
problem Quantum aspects of gauging continuous 1-form global symmetries.
method BV-BRST quantization of a U(1) 2-form gauge field, Lie 2-algebroid construction, Čech-de Rham bicomplex.
result Anomaly descent for U(1) 1-form symmetries is naturally set up in the Čech-de Rham bicomplex.
The paper describes hyperkähler geometry of cotangent bundles using rank-1 projections.
problem Understanding hyperkähler geometry of cotangent bundles via algebraic methods.
method Algebraic description via the scheme of rank-1 projections, isometric embeddings, and generalizations.
result Explicit isometric embeddings and generalizations of hyperkähler geometry.
Study on Čech-de Rham obstruction in diffeological spaces.
problem Obstruction to Čech-de Rham map being an isomorphism in diffeological spaces.
method Higher topos theory, homotopy pullback diagrams, Čech-de Rham bicomplex, -stack cohomology.
result New exact sequences in all higher degrees and conceptual proof of cohomology agreement.
Study quantum aspects of 1-form symmetries using BV-BRST cohomology and gerbes.
problem Quantum aspects of gauging continuous 1-form global symmetries.
method BV-BRST quantization of a U(1) 2-form gauge field, Lie 2-algebroid construction, Čech-de Rham bicomplex.
result Anomaly descent for U(1) 1-form symmetries is naturally set up in the Čech-de Rham bicomplex.
The paper explores non-Kähler SYZ mirrors for solvmanifolds, proving cohomological properties and constructing new mirror pairs.
problem Understanding non-Kähler SYZ mirrors for solvmanifolds and their cohomological properties.
method Investigates geometric and cohomological properties, proving relationships and constructing mirror pairs.
result Proves the Fourier-Mukai transform exchanges type-A and type-B cycles, and provides criteria for non-Kähler SYZ mirror pairs.
Derives Lagrangian for minimal surfaces, proving tangential variations vanish.
problem Variational calculus for minimal surfaces.
method Lagrangian formulation, pullback covariant derivative, geometric argument.
result Tangential variations vanish for minimal surfaces.
Improved Bayesian uncertainty quantification using variational bagging.
problem Inefficient and underestimating uncertainty in mean-field variational Bayes.
method Integrates bagging with variational Bayes for improved inference.
result Bagged variational posterior provides proper uncertainty quantification.
This work improves variational inference by reducing gradient variance.
problem Hard optimization of flexible variational distributions.
method Control variate based on quadratic approximation of the model's mean and covariance.
result Significant improvement in gradient variance and optimization convergence.
Adaptive variational Bayes framework improves inference adaptively.
problem Lack of general and computationally tractable variational Bayes method for adaptive inference.
method Proposes a novel adaptive variational Bayes framework combining variational posteriors over individual models.
result Adaptive variational Bayes achieves optimal contraction rates adaptively under general conditions.
A new EVI framework improves ParVI methods by maintaining variational structure and reducing KL-divergence.
problem Improving variational inference methods for better approximation of target distributions.
method EVI framework that minimizes the VI objective function based on an energy-dissipation law, including a new 'Approximation-then-Variation' scheme.
result The new scheme significantly decreases KL-divergence and outperforms existing ParVI methods in fidelity.
We propose a family of variational approximations to Bayesian posterior distributions, called -VB, with provable statistical guarantees. The standard variational approximation is a special case of -VB with . When , a novel class of variational inequalities are developed for linking the Bayes risk …
On variational bivectorsmath.DG
New examples of variational bivectors found that are not Poissonian.
problem Identifying variational bivectors that are not Poissonian.
method Constructing examples of variational bivectors.
result Found examples of variational bivectors that are not Poissonian.