Study Poisson structures on fibered 5-manifolds with compatibility conditions.
problem Understanding Poisson structures on fibered 5-manifolds.
method Using almost coupling condition and bigraded factorization of the Jacobi identity.
result Describe global behavior and singularities of almost coupling Poisson tensors.
Conditions for statistical structures on manifolds derived from solitons.
problem Characterizing statistical structures on manifolds from soliton equations.
method Analyzing gradient solitons on statistical manifolds to derive conditions for statistical structures.
result Established necessary and sufficient conditions for statistical structures under various soliton types.
BGG-equations are geometric overdetermined systems of PDEs on parabolic geometries. Normal solutions of BGG-equations are particularly interesting and we give a simple formula for the necessary and sufficient additional integrability conditions on a solution. We then discuss a procedure for coupling known solutions of …
We show that the properties of Lagrangian mean curvature flow are a special case of a more general phenomenon, concerning couplings between geometric flows of the ambient space and of totally real submanifolds. Both flows are driven by ambient Ricci curvature or, in the non-Kähler case, by its analogues. To this end we…
Study finds almost contact structures in thermal QCD-like theories at intermediate coupling.
problem Understanding (Almost) Contact Structures in thermal QCD-like theories.
method Explicitly obtained (Almost) Contact Structures and SU(3) structures.
result Subspaces of C3S and AC3S are not mutually 'N-path connected' in the Infra-Red.
We define a non-degenerated Monge-Ampere structure on a 6-manifold associated with a Monge-Ampere equation as a couple (Ω,ω), such that Ωis a symplectic form and ωis a 3-differential form which satisfies ω\wedgeΩ=0 and which is non-degenerated in the sense of Hitchin. We associate with such a couple an almost (pseudo) …
First-order methods play a central role in large-scale machine learning. Even though many variations exist, each suited to a particular problem, almost all such methods fundamentally rely on two types of algorithmic steps: gradient descent, which yields primal progress, and mirror descent, which yields dual progress. W…
The paper explores how a geometric flow can turn a black hole into a traversable wormhole.
problem The study investigates how a static, spherically symmetric black hole can be transformed into a traversable wormhole.
method The approach involves analyzing almost η-Ricci-Yamabe solitons and their geometric coupling with the Hawking temperature. result The geometric flow successfully transforms the black hole into a traversable wormhole, opening the throat and preserving the exact cosmological spacetime.
Paper discusses conditions for deforming coupled Kähler-Einstein metrics.
problem Conditions for deforming coupled Kähler-Einstein metrics.
method Analyzes deformation of coupled Kähler-Einstein metrics on Fano manifolds.
result Necessary and sufficient condition for deformation of coupled Kähler-Einstein metrics.
This paper shows how to approximate any log-concave distribution using well-conditioned affine coupling flows.
problem Understanding the representational power of affine coupling flows for log-concave distributions.
method Leveraging connections between affine coupling architectures, Langevin dynamics, and Hénon maps to prove log-concave approximation.
result Any log-concave distribution can be approximated using well-conditioned affine-coupling flows.
Unified Jacobi coupling construction for various geometric settings.
problem Constructing Jacobi structures on associated bundles.
method Extending Sternberg--Weinstein coupling to Jacobi geometry.
result Associated bundles inherit Jacobi structures from base spaces.
Paper explores existence and nonexistence of submanifolds in contact geometry.
problem Existence and nonexistence of warped product submanifolds in almost contact manifolds.
method Generalizes results for existence and nonexistence, constructs examples, addresses open problems.
result Offers decisive results for existence and nonexistence of submanifolds.
Numerical observations on martingale couplings are confirmed under certain conditions.
problem Understanding the validity of numerical observations on maximizers and minimizers of martingale couplings.
method Investigation of sufficient conditions and counterexamples for the property to hold.
result The non-decreasing property of martingale couplings is preserved for maximizers under specific conditions.
A new method for conditional sampling using paired Wasserstein Autoencoders.
problem Conditional sampling from complex data distributions.
method Derive a novel loss function for Wasserstein Autoencoders to enable sampling from OT-type couplings.
result Learned cost-optimal transport maps and conditional sampling from an OT-type coupling.
Develop a framework for barycentric projections of optimal transport plans on Riemannian manifolds.
problem Optimal transport couplings are probabilistic objects, while many learning pipelines require deterministic maps.
method Develop a framework for barycentric projections of transport couplings on Riemannian manifolds.
result The intrinsic projection maps each source point to the conditional Fréchet mean of its destination law and is shown to be the best deterministic representative under squared geodesic loss.
Conditions for Penrose-Ward transformation on specific manifolds.
problem Conditions for Penrose-Ward transformation on almost G2-manifolds with almost twistorial structures. method Necessary and sufficient conditions derived through Penrose-Ward transformation.
result Conditions for Penrose-Ward transformation on almost G2-manifolds with almost twistorial structures. For a finite family of 3-dimensional almost contact metric manifolds with closed the structure form η is described a construction of an almost contact metric manifold, where the members of the family are building blocks - cells. Obtained manifold share many properties of cells. One of the more important are nullity c…
We review coupled SU(3)-structures, also known in the literature as restricted half-flat structures, in relation to supersymmetry. In particular, we study special classes of examples admitting such structures and the behaviour of flows of SU(3)-structures with respect to the coupled condition.
The study proves conditions for Hermitian metrics on compact almost complex manifolds.
problem Conditions for Hermitian metrics on compact almost complex manifolds.
method Analyzes compact almost complex manifolds with Hermitian metrics and integral conditions involving ∂-harmonic (0,1)-forms. result The integral condition is automatically satisfied for strongly Gauduchon metrics, and equivalent to being strongly Gauduchon for integrable almost complex structures.
We prove that minimal graphs (other than planes) are parabolic in the sense that any bounded harmonic function is determined by its boundary values. The proof relies on using the coupling introduced in the author's earlier paper "A martingale approach to minimal surfaces" to show that Brownian motion on such a minimal …
Two probability distributions μ and ν in second stochastic order can be coupled by a supermartingale, and in fact by many. Is there a canonical choice? We construct and investigate two couplings which arise as optimizers for constrained Monge-Kantorovich optimal transport problems where only supermartingales are al…
We consider the semi-classical Dirac operator coupled to a magnetic potential on a large class of manifolds including all metric contact manifolds. We prove a sharp local Weyl law and a bound on its eta invariant. In the absence of a Fourier integral parametrix, the method relies on the use of almost analytic continuat…
A new neural network architecture reduces parameters and improves performance.
problem Reducing model complexity and improving performance in neural networks.
method Coupled ensembles of neural networks with parallel branches and tighter coupling.
result Significant performance improvements with reduced parameter count.
Characterizes a class of almost Hermitian 4-manifolds using integral identities.
problem Global characterization of almost Hermitian 4-manifolds.
method Using an integral identity from Sekigawa's work, proving a uniqueness result on Lie algebras.
result Global characterization of the class AH1 of almost Hermitian 4-manifolds. Proof confirms condition for Kähler-Einstein metrics on toric Fano manifolds.
problem Existence of Kähler-Einstein metrics on toric Fano manifolds.
method Condition in terms of barycenters of polytopes.
result Necessary and sufficient conditions for existence of coupled Kähler-Einstein metrics and soliton solutions.
In this work, almost product and almost golden structures are studied. Conditions for those structures being Integrable and parallel are investigated. Also harmonicity of a map between almost pruduct or almost golden manifolds with pure or hyperbolic metric is discussed under certain conditions.
The paper explores Hodge decomposition and Hard Lefschetz Condition on almost Kähler manifolds.
problem Analyzing harmonic forms and Hodge decomposition on almost Kähler manifolds.
method Using Hodge decomposition and the Hard Lefschetz Condition to study almost Kähler manifolds.
result The spaces of harmonic forms have the Hodge decomposition and the Hard Lefschetz Condition is satisfied.
We show that the left-monotone martingale coupling is optimal for any given performance function satisfying the martingale version of the Spence-Mirrlees condition, without assuming additional structural conditions on the marginals. We also give a new interpretation of the left monotone coupling in terms of Skorokhod e…
A locally conformally symplectic (LCS) form is an almost symplectic form ω such that a closed one-form θ exists with dω=θ∧ω. A fiber bundle with LCS fiber (F,ω,θ) is called LCS if the transition maps are diffeomorphisms of F preserving ω (and hence θ). In this paper, we find conditions for the total…
Estimates eigenvalue for manifolds with specific forms under certain conditions.
problem Estimating the first eigenvalue of Laplacian on manifolds with almost parallel p-forms. method Uses Lichnerowicz-Obata estimate and pinching conditions to analyze eigenvalues.
result Establishes a Lichnerowicz-Obata type estimate for the first eigenvalue.
Extends GP regression to complex Helmholtz problems, improving wavefield inference in brain elastography.
problem Infer complex Helmholtz wavefields from sparse, noisy data.
method Operator-informed Gaussian processes, realifying complex operator into real blocks, using PDE residuals and boundary traces.
result Competitive with finite-difference and neural-network methods, reconstructs brain shear curl field with high correlation.
This paper is a complete study of almost α-paracosmplectic manifolds. We characterize almost α-paracosmplectic manifolds which have para Kaehler leaves. Main curvature identities which are fulfilled by any almost α-paracosmplectic manifold are found. We also proved that ξ is a harmonic vector field if and only if it is…
We propose new types of canonical metrics on Kähler manifolds, called coupled Kähler-Einstein metrics, generalizing Kähler-Einstein metrics. We prove existence and uniqueness results in the cases when the canonical bundle is ample and when the manifold is Kähler-Einstein Fano. In the Fano case we also prove that existe…
Almost-Riemannian manifolds fail to meet a synthetic curvature condition.
problem Proving almost-Riemannian manifolds do not satisfy the CD condition. method Developed a new strategy to contradict the 1-dimensional CD condition. result 2D and strongly regular almost-Riemannian manifolds do not satisfy CD(K,N) for any K and N. A new flow family reduces to Li-Yuan-Zhang's and can lead to convergence under certain conditions.
problem Establishing convergence of coupled flow equations under various conditions.
method Introducing a one-parameter family of coupled flows and applying C0 estimates and monotonicity of energy functionals. result Convergence of the flow can be established for κeq1 under suitable conditions. We construct supersymmetric field theories on Riemannian three-manifolds M, focusing on N=2 theories with a U(1)_R symmetry. Our approach is based on the rigid limit of new minimal supergravity in three dimensions, which couples to the flat-space supermultiplet containing the R-current and the energy-momentum tensor. T…
The paper investigates geometrical aspects of static spacetime with almost gradient Ricci solitons.
problem Geometrical properties of static spacetime with almost gradient Ricci solitons.
method Analyzing conditions and properties of static spacetime with almost gradient Ricci solitons.
result Conditions and properties of static spacetime with almost gradient Ricci solitons are determined.
The paper examines conditions for a vector field to be Killing in almost Yamabe solitons.
problem Conditions for a vector field to be Killing in almost Yamabe solitons.
method Investigation of almost Yamabe solitons on compact and non-compact manifolds.
result Sufficient conditions for the defining conformal vector field to be Killing.
Study finds optimal martingale coupling between two distributions with minimal entropy.
problem Finding the optimal martingale coupling between two distributions with minimal relative entropy.
method Solving a dual problem to find the log-density of the optimal coupling, which represents the marginal and martingale constraints.
result The log-density of the optimal coupling is given by a triplet of real functions representing the marginal and martingale constraints.
ReDi improves few-step generation for discrete data models.
problem Slow sampling speeds in discrete flow-based models.
method Rectified Discrete Flow (ReDi) reduces factorization error by rectifying coupling.
result Empirically, ReDi reduces Conditional Total Correlation and enables few-step generation.
New framework explains normalizing flows' power and limitations.
problem Understanding the expressive power and limitations of normalizing flows.
method Theoretical framework for well-conditioned coupling-based normalizing flows and volume-preserving flows.
result RealNVP is distributionally universal, but volume-preserving flows are not.
Conditions for solutions to complex Monge-Ampère equations on Fano manifolds.
problem Existence of solutions to complex Monge-Ampère equations on Fano horosymmetric manifolds.
method Necessary and sufficient conditions derived from combinatorial data.
result Conditions for existence of solutions in terms of combinatorial data.
New theorem using Ricci flow for Gromov almost flat manifolds.
problem Conditions for Gromov almost flat manifolds.
method Employing Ricci flow to derive a new theorem.
result New theorem with weaker condition than Gromov--Ruh Theorem.
Paper shows examples of almost Kähler manifolds satisfying Hard Lefschetz but not Betti-Hodge equality.
problem Understanding the Hard Lefschetz condition in almost Kähler manifolds.
method Examples and counterexamples of compact almost Kähler manifolds.
result The Hard Lefschetz condition does not imply the equality between Betti and Hodge numbers in almost Kähler manifolds.
A coupling by reflection of a time-inhomogeneous diffusion process on a manifold are studied. The condition we assume is a natural time-inhomogeneous extension of lower Ricci curvature bounds. In particular, it includes the case of backward Ricci flow. As in time-homogeneous cases, our coupling provides a gradient esti…
We give sufficient conditions for the existence of a Dirac structure on the total space of a Poisson fiber bundle endowed with a compatible connection. We also show that Cartan and Cartan-Hannay-Berry connections give rise to coupling Dirac structures.
We give a necessary and suffcient condition for almost-flat manifolds with cyclic holonomy to admit a Spin structure. Using this condition we find all 4-dimensional orientable almost- flat manifolds with cyclic holonomy that do not admit a Spin structure.
Study new submersion types on quaternionic manifolds.
problem Explore new submersion types on quaternionic manifolds.
method Introduce and study h-conformal semi-invariant submersions and almost h-conformal semi-invariant submersions.
result Properties of these submersions, including foliations and geodesic conditions.