This paper shows the reduced characteristic group of Lie LCP manifolds is simply connected.
problem Understanding the structure of Lie LCP manifolds and their characteristic groups.
method Restricting the action of the fundamental group to the non-flat factor of the universal cover and taking the connected component of the identity in the closure of this restriction.
result The reduced characteristic group of any Lie LCP manifold is simply connected.
New manifold types defined on quotient spaces.
problem Characterizing specific manifold structures.
method Introducing decomposable LCP manifolds and studying their properties.
result Characterized LCP manifolds on quotient spaces.
Locally conformally product Lie algebras are characterized and constructed.
problem Characterizing and constructing compact locally conformally product Lie algebras.
method Characterization via closed 1-forms and non-unimodular Lie algebras acting on abelian ones.
result Explicit examples of compact LCP manifolds that are not solvmanifolds.
This paper describes the characteristic group of LCP structures and their construction.
problem Understanding the characteristic group of LCP structures and their construction.
method Analyzing the characteristic group and constructing LCP structures.
result It is possible to endow certain quotients with LCP structures.
Study LCP structures on solvmanifolds, complete list up to 5 dimensions.
problem Characterize LCP structures on solvable Lie groups.
method Classify LCP structures on Lie groups, focus on solvable unimodular case.
result Complete list of solvable unimodular Lie algebras up to dimension 5 with LCP structures.
Conformal vector fields on LCP manifolds are orthogonal and Killing.
problem Understanding conformal vector fields on specific geometric manifolds.
method Analyzing properties of conformal vector fields on compact locally conformally product manifolds.
result Conformal vector fields are orthogonal to the flat distribution and Killing.
Locally conformally product structures defined on compact manifolds.
problem Defining and characterizing locally conformally product structures on compact manifolds.
method Using reducible, non-flat, incomplete Riemannian metrics and connections with specific properties.
result Construction of new LCP structures from existing ones and links to number field theory.
The abstract discusses compact quotients of Riemannian products by discrete subgroups, generalizing Inoue-Bombieri surfaces.
problem Compact quotients of Riemannian products by discrete subgroups.
method Study of compact quotients of a Riemannian product Rqimes(N,gN) by discrete subgroups Γ of Sim(Rq)imesIsom(N). result The construction is equivalent to LCP manifolds and provides a Bieberbach-type rigidity result.
In this paper, we present a novel unsupervised feature learning architecture, which consists of a multi-clustering integration module and a variant of RBM termed multi-clustering integration RBM (MIRBM). In the multi-clustering integration module, we apply three unsupervised K-means, affinity propagation and spectral c…
New projection techniques reduce the frequency of projections in solving LCPs.
problem Solving linearly constrained problems efficiently with reduced projection frequency.
method Delayed projection technique to call a projection less frequently.
result Theoretical and practical improvements in convergence rates and efficiency.
Classifies Weyl structures on compact conformal manifolds with special holonomy.
problem Classifying Weyl structures on compact conformal manifolds with specific holonomy properties.
method Analyzes the properties of connections preserving the conformal structure and classifies structures based on their holonomy.
result Local classification of Weyl structures on compact conformal manifolds with special holonomy.
Paper proposes LCP for structural encodings, outperforming existing methods.
problem Improving Graph Neural Networks performance through effective structural encodings.
method Geometric perspective, Local Curvature Profiles (LCP) for structural encodings, combining with global positional encodings, comparing with rewiring techniques.
result LCP significantly outperforms existing structural encodings and combining LCP with global positional encodings improves performance.
European options can be priced by solving parabolic partial(-integro) differential equations under stochastic volatility and jump-diffusion models like Heston, Merton, and Bates models. American option prices can be obtained by solving linear complementary problems (LCPs) with the same operators. A finite difference di…
In this paper, we demonstrate that policy iteration, introduced in the context of HJB equations in [Forsyth & Labahn, 2007], is an extremely simple generic algorithm for solving linear complementarity problems resulting from the finite difference and finite element approximation of American options. We show that, in ge…
Positive injectivity radius for manifolds with Lie structure at infinity.
problem Injectivity radius positivity for manifolds with specific boundary conditions.
method Lie groupoids to prove injectivity radius positivity.
result Injectivity radius is positive for manifolds with Lie structure at infinity.
Reduces Poisson manifolds with Hamiltonian Lie algebroids.
problem Handling Poisson manifolds with Hamiltonian Lie algebroids.
method Introducing compatibility of momentum sections and quotienting zero level sets.
result Quotient space of zero level set of compatible momentum section is a Poisson manifold.
Study connects Lie groups to specific Riemannian manifolds.
problem Understanding Lie groups through Riemannian manifold properties.
method Investigates Lie groups as 3D almost paracontact almost paracomplex Riemannian manifolds.
result Established correspondence between Lie algebra and matrix representation.
Study on 4D Lie groups and related almost hypercomplex manifolds.
problem Characterizing almost hypercomplex manifolds with specific metrics.
method Construction and classification of manifolds based on Lie algebras.
result Established a connection between Lie algebra classes and manifold classifications.
Investigate local Lie group structure of bisections over compact manifolds
problem Study the local Lie group structure associated with the space of admissible bisections of a local Lie groupoid over a compact manifold.
method Investigate the relation of this local Lie group to the Lie algebra of sections of the associated Lie algebroid.
result Prove that the globalizability of a local Lie groupoid implies the globalizability of its associated local Lie group of bisections.
Integrable LCK manifolds characterized as Kähler Lie algebras.
problem Characterizing LCK manifolds with integrable anti-Lee forms.
method Examining LCK manifolds with integrable anti-Lee forms and applying to Lie algebras.
result Unimodular integrable LCK Lie algebras are Kähler Lie algebras with specific derivations.
Study finds specific Lie groups with Kenmotsu structures.
problem Characterizing Lie groups with Kenmotsu structures.
method Determined Lie groups with left invariant Kenmotsu structures.
result These Lie groups are Einstein Riemannian manifolds.
Statistical Lie algebras with constant curvature are linked to locally conformally Kähler structures.
problem Characterizing Lie algebras with constant curvature and their geometric implications.
method Constructing statistical manifolds and Sasakian structures to relate Lie algebras to locally conformally Kähler structures.
result Statistical Lie algebras of constant curvature correspond to locally conformally Kähler Lie algebras.
Study Hom-Lie algebroid connections on complex manifolds.
problem Irreducible connections on Hom-Lie algebroids.
method Proved moduli space structure using H-gauge theory.
result Moduli space has a Hausdorff Hilbert manifold structure.
The notion of Poisson manifold with compatible pseudo-metric was introduced by the author in [1]. In this paper, we introduce a new class of Lie algebras which we call a pseudo-Rieamannian Lie algebras. The two notions are strongly related: we prove that a linear Poisson structure on the dual of a Lie algebra has a com…
Results on characterization of manifolds in terms of certain Lie algebras growing on them, especially Lie algebras of differential operators, are reviewed and extended. In particular, we prove that a smooth (real-analytic, Stein) manifold is characterized by the corresponding Lie algebra of linear differential operator…
New Q-manifolds theory integrates Lie algebroids.
problem Integrating Lie algebroids over smooth manifolds.
method Introducing Q-groupoids and Q-bundles, proving Lie algebroids arise from Q-manifolds.
result Transitive Lie algebroids over second countable, smooth manifolds are integrated to locally trivial Q-groupoids.
Introduces comomentum sections and proves they are Poisson maps.
problem Generalizing Poisson maps to Hamiltonian Lie algebroids.
method Introduces comomentum sections and proves they are Lie algebroid morphisms and Poisson maps.
result Comomentum sections are Poisson maps between proper Poisson manifolds.
Study Einstein Lie groups and geodesic orbit manifolds, finding some are not geodesic orbit.
problem Characterizing Einstein Lie groups and geodesic orbit manifolds.
method Characterizing GimesK-invariant geodesic orbit metrics on Lie groups G for regular subgroups K. result Extensive classes of compact simple Einstein Lie groups are not geodesic orbit manifolds.
Study of unimodular Sasaki and Vaisman Lie groups, determining all modifications explicitly.
problem Classifying unimodular Sasaki and Vaisman Lie groups.
method Applying the technique of modification to determine all homogeneous Sasaki and Vaisman manifolds of unimodular Lie groups explicitly.
result Complete classification of unimodular Sasaki and Vaisman Lie groups.
Develops a spectral sequence for Lie group actions on manifolds.
problem Understanding cohomology of manifolds with Lie group actions.
method Introduces a spectral sequence relating manifold cohomology to Lie algebra cohomology.
result Establishes a new description of de Rham cohomology for manifolds with Lie group actions.
Extends Cheeger's method to Lie groupoid actions on manifolds.
problem Smooth Lie group actions on manifolds with singularities.
method Extension of Cheeger's deformation techniques to Lie groupoid actions.
result Explicit sectional curvature description of the deformation.
We define Lie and Courant algebroids on Fréchet manifolds. Moreover, we construct a Dirac structure on the generalized tangent bundle of a Fréchet manifold and show that it inherits a Fréchet Lie algebroid structure. We show that the Lie algebroid cohomology of the $\bb$-cotangent bundle Lie algebroid of a weakly sympl…
New Lie systems defined on k-contact manifolds, with applications.
problem Understanding Lie systems on k-contact manifolds. method Distributional approach to k-contact manifolds and Hamiltonian vector fields. result Lie systems can be understood as Hamiltonian relative to a k-contact manifold. Defines and extends Lie algebroid prolongations in convenient settings.
problem Adapting Lie algebroid prolongations to convenient settings.
method Defined and adapted Lie algebroid prolongations over fibred manifolds.
result Stability of prolongations under projective and direct limits.
In this paper, first we modify the definition of a Hom-Lie algebroid introduced by Laurent-Gengoux and Teles and give its equivalent dual description. Many results that parallel to Lie algebroids are given. In particular, we give the notion of a Hom-Poisson manifold and show that there is a Hom-Lie algebroid structure …
Local Kan conditions enable differentiation of simplicial manifolds.
problem Differentiating simplicial manifolds into Lie algebroids.
method Expanding a technique for higher Lie groupoids to simplicial manifolds.
result Derivation of a method to differentiate simplicial manifolds into higher Lie algebroids.
Paper introduces Lie algebroid index theory and a generalized Riemann-Roch theorem.
problem Generalizing Riemann-Roch theorem for manifolds with regular foliations.
method Developed Lie algebroid index theory and applied it to obtain a generalized Riemann-Roch theorem.
result Obtained a generalized Riemann-Roch theorem for manifolds with regular foliations.
Study weightings from singular Lie filtrations.
problem Generalize constructions for singular Lie filtrations.
method Study weightings arising from singular Lie filtrations.
result Generalizes constructions for (regular) Lie filtrations.
Lie PCA improves density estimation on symmetric manifolds.
problem Density estimation for symmetric manifolds.
method Spectral method to approximate Lie algebra of symmetry group.
result Improved sample complexity and density estimation on various data sets.
A manifold with a ``Lie structure at infinity'' is a non-compact manifold M0 whose geometry is described by a compactification to a manifold with corners M and a Lie algebra of vector fields on M, subject to constraints only on M∖M0. The Lie structure at infinity on M0 determines a metric on $M_…
New Lie groups found for Poisson diffeomorphisms.
problem Finding Lie group structures on Poisson diffeomorphism groups.
method Using Poisson groupoids, develop Lie group structures.
result Poisson diffeomorphism groups of various Poisson manifolds are regular Lie groups.
Discuss various definitions of vector fields on manifolds leading to Lie algebras.
problem Defining vector fields on convenient manifolds.
method Various definitions of vector fields and their equivalence.
result Equivalent definitions of vector fields on manifolds.
Develops Lie algebraic approach for compact complex homogeneous manifolds.
problem Proves important results on compact complex homogeneous manifolds.
method Uses standard results in Lie theory to associate a canonical abelian Lie algebra with a given integrable complex structure.
result Provides a new method of associating a canonical abelian Lie algebra with a given integrable complex structure.
Study examines Lie algebroids with homological sections, generalizing Q-manifolds and Lie superalgebras.
problem Exploring Lie algebroids with homological sections.
method Derived bracket formalism to define an odd Loday-Leibniz bracket on sections.
result Sections of inner Q-algebroids come equipped with an odd Loday-Leibniz bracket.
Characterizes connections on normal distributions manifold.
problem Geometric characterization of connections on normal distributions.
method Homogeneous statistical manifold structure and Lie group analysis.
result Geometric characterization of α-connections on Lie group. The paper introduces a new form on Lie algebroids over multisymplectic manifolds.
problem Higher generalizations of Poisson structures and momentum maps.
method Introducing a compatible E-n-form on Lie algebroids.
result The introduced form satisfies a compatibility condition with Lie algebroid and multisymplectic structures.
We introduce a new cohomology for Lie algebroids, and prove that it provides a differential graded Lie algebra which ``controls'' deformations of the structure bracket of the algebroid. We also have a closer look at various special cases such as Lie algebras, Poisson manifolds, foliations, Lie algebra actions on manifo…
The object of investigation are Lie groups considered as almost contact B-metric manifolds of the lowest dimension three. It is established a correspondence of all basic-class-manifolds of the Ganchev-Mihova-Gribachev classification of the studied manifolds and the explicit matrix representation of Lie groups. Some kno…