Study on the Kodaira dimension of real parallelizable manifolds with almost complex structures.
problem Understanding the Kodaira dimension of real parallelizable manifolds with specific almost complex structures.
method Conditions and examples provided for calculating the Kodaira dimension of manifolds.
result Conditions under which the Kodaira dimension of a real parallelizable manifold is zero.
Regularities and stability shown for a specific type of complex parallelizable manifolds.
problem Stability and regularity of Chern-flat metrics on complex parallelizable manifolds.
method Study of Hermitian metrics governed by the second Chern-Ricci form on compact complex manifolds.
result Chern-flat metrics are dynamically stable on compact complex parallelizable manifolds.
New findings on Kähler structures on specific types of manifolds.
problem Classifying manifolds with p-Kähler structures. method Analyzing nilmanifolds and holomorphically parallelizable solvmanifolds.
result Classification of manifolds with p-Kähler structures for lower values of p. Three proofs show all 3D shapes are parallelizable without complex tools.
problem Proving all 3D shapes are parallelizable without advanced tools.
method Minimal proofs using no spin structures or Stiefel-Whitney classes.
result Three proofs show all 3D shapes are parallelizable.
Study the rank of Nijenhuis tensor on parallelizable almost complex manifolds.
problem Understanding the rank of Nijenhuis tensor on parallelizable almost complex manifolds.
method Reduction of computations to solving PDEs, explicit solutions on specific manifolds, analysis of curve of almost complex structures, classification of Lie algebras.
result Classification of Lie algebras admitting almost complex structures with specific Nijenhuis tensor ranks.
Stability of flat vector bundles on complex parallelizable manifolds proven.
problem Stability of flat holomorphic vector bundles over compact complex parallelizable manifolds.
method Proved a structure theorem for flat holomorphic vector bundles associated to irreducible representations.
result Flat holomorphic vector bundles are holomorphically isomorphic to a stable vector bundle of the form E⊕n. A 4-manifold is parallelizable if its Stiefel-Whitney and Pontryagin classes vanish.
problem Characterizing parallelizable 4-manifolds.
method Classification of SO(4)-bundles over the 4-sphere using Euler and first Pontryagin classes. result A closed orientable 4-manifold is parallelizable if and only if its second Stiefel-Whitney class, first Pontryagin class, and Euler characteristic vanish.
Study on special metrics and deformations of solvmanifolds.
problem Existence and properties of Kähler metrics on solvmanifolds.
method Investigation of strong Kähler with torsion metrics and balanced metrics on deformations of specific solvmanifolds.
result Non-existence of certain metrics on specific solvmanifolds.
Proof of orientable 3-manifolds parallelizability using knot theory.
problem Proving all orientable 3-manifolds are parallelizable.
method Using knot theory and relationships between tangent and normal bundles.
result Completion and modification of a proof of Stiefel's theorem.
Study on almost complex structures with maximal Nijenhuis tensor rank and cohomological properties.
problem Maximally non-integrable almost complex structures and their cohomological properties.
method h-principle and topological invariants characterization.
result Existence of almost complex structures with maximal Nijenhuis tensor rank on parallelizable and certain manifolds.
The paper explores p-Kähler structures on complex manifolds and their properties.
problem Addressing conjectures on specific types of complex manifolds and their structures.
method Analyzing (n−2)-Kähler nilmanifolds and holomorphically parallelizable nilmanifolds, deriving necessary conditions for smooth curves of p-Kähler structures, studying cohomology classes, and providing examples. result Derives necessary conditions for the existence of smooth curves of p-Kähler structures and provides examples of compact complex manifolds with p-Kähler structures. We classify complex compact parallelizable manifolds which admit flat torsion free holomorphic affine connections. We exhibit complex compact manifolds admitting holomorphic affine connections, but no flat torsion free holomorphic affine connections.
New proof shows 3D shapes can be continuously deformed.
problem Proving 3D shapes can be continuously deformed without tearing.
method Used Heegaard splitting as a main tool.
result Closed orientable 3-manifolds are parallelizable.
We study conditions under which sub-complexes of a double complex of vector spaces allow to compute the Bott-Chern cohomology. We are especially aimed at studying the Bott-Chern cohomology of special classes of solvmanifolds, namely, complex parallelizable solvmanifolds and solvmanifolds of splitting type. More precise…
We prove a stability theorem for families of holomorphically-parallelizable manifolds in the category of Hermitian manifolds.
Computes Picard groups of complex parallelizable manifolds.
problem Calculating Picard groups of specific compact complex manifolds.
method Analyzes tangent bundle triviality and uses lattice and Lie group properties.
result Computes Picard groups for certain compact complex parallelizable manifolds.
The paper proves conditions for 4-manifolds to be parallelizable and non-existence results.
problem Conditions for 4-manifolds to be parallelizable and non-existence results.
method Homotopy theory, particularly obstruction theory.
result Compact, oriented 4-manifolds with vanishing Euler characteristic and signature admit coassociative-free immersions.
The paper explores algebraic and geometric structures on parallelizable manifolds.
problem Understanding algebraic and geometric structures on parallelizable manifolds.
method Definition of fundamental vector fields and their flows, leading to a product and loop structure.
result Induces a local loop structure and generalizes Lie algebra structure on the vector space.
Study homogenizes equations on parallelizable manifolds using tensor localization and periodicity.
problem Homogenizing oscillating linear elliptic equations on parallelizable manifolds.
method Two-scale convergence through localization and periodicity induced by geometry.
result Explicit cell formulae for the homogenization limit and a theory of two-scale convergence of tensors.
A holomorphic Poisson structure induces a deformation of the complex structure as Hitchin's generalized geometry. Its associated cohomology naturally appears as the limit of a spectral sequence of a double complex. The first sheet of this spectral sequence is the Dolbeault cohomology with coefficients in the exterior a…
Motivated by the Hamilton's Ricci flow, we define the homogeneous flow of a parallelizable manifold and show the long time existence and uniqueness of its solutions on [0,∞). Using this flow, we outline a simple proof of the Poincare Conjecture.
New complex non-Kähler manifolds with specific properties are constructed.
problem Constructing complex non-Kähler manifolds with special properties.
method Using families of compact solvmanifolds and properties of the ∂∂ˉ-Lemma. result Provided families of compact (n+1)-dimensional complex non-Kähler manifolds with specific properties. New 4D shapes can't be opened like books.
problem Existence of 4D shapes without open book decompositions.
method Demonstrated existence of infinitely many parallelizable 4-manifolds without open book decompositions.
result No open book decomposition for certain Engel manifolds.
Study of deformed Bott-Chern cohomology on complex manifolds.
problem Deformation theory and cohomology of complex manifolds.
method Introduce a double complex structure and study its Bott-Chern cohomology.
result Established a deformation theory for Bott-Chern cohomology and computed deformed cohomology for specific manifolds.
We show that if M is an orientable 4-dimensional infrasolvmanifold and either β=β1(M;Q)≥2 or M is a Sol04- or a Solm,n4-manifold (with m=n) then M is parallelizable. There are non-parallelizable examples with β=1 for each of the other solvable Lie geometries $\ma…
Study finds infinite families of Sasaki-Einstein metrics on spheres.
problem Finding Sasaki-Einstein metrics on spheres and exotic spheres.
method Analyzing odd-dimensional spheres and exotic spheres that bound parallelizable manifolds.
result Infinitely many families of Sasaki-Einstein metrics on spheres and exotic spheres.
SpInGP speeds up Gaussian process computations with sparse matrices.
problem Efficiently computing Gaussian processes for large datasets.
method Sparse precision Gaussian process formulation and parallelizable matrix routines.
result The parallelized SpInGP reduces time complexity to sublinear.
Study on existence of balanced metrics on non-Kähler manifolds.
problem Existence of balanced metrics on non-Kähler complex manifolds.
method Analyzes obstructions and constructs examples, focusing on compact quotients of Lie groups.
result Proves non-existence on certain non-Kähler complex parallelizable manifolds and solvmanifolds.
Study sub-Riemannian geometry on parallelizable distributions.
problem Investigate the geometry of sub-Riemannian structures on non-integrable parallelizable distributions.
method Construct and analyze two linear connections: Weitzenböck and sub-Riemannian.
result Two concrete examples (S^3 and S^7) illustrate the application of sub-Riemannian geometry.
Let G be a simply connected solvable Lie group with a lattice Γ and the Lie algebra $\g$ and a representation ρ:G→GL(Vρ) whose restriction on the nilradical is unipotent. Consider the flat bundle Eρ given by ρ. By using "many" characters {α} of G and "many" flat line bundles {Eα} over G/Γ, w…
This article introduces the problem of finding intrinsic torsion varieties associated to G-structures on a fixed parallelizable Riemannian manifold. As an illustration, the intrinsic torsion varieties of orthogonal almost product structures are analysed on the Iwasawa manifold.
We give an elementary proof of the fact that any orientable 3-manifold admits a framing (i.e. is parallelizable) and any non-orientable 3-manifold admits a projective framing. The proof uses only basic facts about immersions of surfaces in 3-space.
Classification of homogeneous almost complex 4-manifolds with non-degenerate torsion bundle
problem Classification of homogeneous almost complex 4-manifolds with non-degenerate torsion bundle
method Classification using rigidity of structures
result Complete classification in the homogeneous setting
The study explores flat connections on manifolds with non-Euclidean signatures.
problem Exploring flat connections on manifolds with non-Euclidean signatures.
method Examined the role of torsion in constructing flat connections and covered Riemannian and pseudo-Riemannian manifolds of dimensions three and seven.
result Explicit flat so(p,q) connections were constructed for coset manifolds SO(p,q+1)/SO(p,q) or SO(p+1,q)/SO(p,q). We prove that no 14-connected (resp. 30-connected) stably parallelizable manifold N30 (resp. N62) of dimension 30 (resp. 62) with the Arf-Kervaire invariant 1 can be smoothly embedded into R36 (resp. R83).
New fault-tolerant quantum gates for homological LDPC codes with constant or almost-constant rate.
problem Fault-tolerant quantum computing for homological LDPC codes with constant or almost-constant encoding rate.
method Derive generic formula for transversal and logical gates acting on 3-manifolds, using higher symmetries and cup product cohomology.
result Parallelizable logical gates for homological LDPC codes with constant or almost-constant rate.
Study on ruled surfaces over elliptic curves with unique foliations and parallelizable 4-webs.
problem Characterizing the geometry of ruled surfaces over elliptic curves.
method Analysis of foliations, minimal self-intersection sections, and 2-webs.
result The 4-web defined by fibration, foliation, and minimal self-intersection sections is locally parallelizable.
Modified BFGS and LBFGS++ libraries boost performance for non-parallelizable functions.
problem Improving performance of non-parallelizable functions using SIMD and AAD.
method Modifications to BFGS and LBFGS++ libraries, utilizing SIMD and Automatic Differentiation (AAD).
result Up to 3.8 times faster for European Swaption curve calibration and 1.4 times faster for LMM model calibration.
Identifies special Lagrangian submanifolds in non-Kähler Calabi-Yau manifolds.
problem Determining special Lagrangian submanifolds in non-Kähler Calabi-Yau manifolds.
method Algebraic equations, invariant distributions, deformation theory, SYZ mirror symmetry.
result Existence of topologically distinct SLags and non-Kähler SYZ mirrors.
We develop a construction of Engel stuctures on 4-manifolds based on decompositions of manifolds into round handles. This allows us to show that all parallelizable 4-manifolds admit an Engel structure. We also show that, given two Engel manifolds M_1,M_2 satisfying a certain condition on the characteristic foliation, t…
New CNN model for fast unsupervised document embedding.
problem Efficient unsupervised document embedding with parallelizable architecture.
method Convolutional Neural Network (CNN) for parallel inference and stochastic forward prediction for unsupervised learning.
result Comparable accuracy to state-of-the-art at significantly reduced computational cost.
Study proves a space of manifolds is homology equivalent to a loop space.
problem Understanding the stable moduli space of certain high-dimensional manifolds.
method Incorporates surgery data via Lagrangian subspaces in a cobordism category.
result The stable moduli space is homology equivalent to an infinite loopspace.
Study on Kuranishi spaces of complex structures and vector bundles, showing isomorphisms and counterexamples.
problem Understanding Kuranishi spaces of complex structures and vector bundles.
method Analyzing Kuranishi spaces of pairs (M,E) of compact Kähler manifolds and vector bundles. result Isomorphisms and counterexamples of Kuranishi spaces of pairs (M,E) of nilmanifolds and trivial vector bundles. Proposes SVM-based Deep Stacking Network for improved deep learning.
problem Improving deep learning performance and interpretability.
method Uses stacked SVM classifiers within a DSN architecture and a BP-like layer tuning scheme.
result Demonstrates superior performance compared to benchmark models on image and text data.
Stochastic Variational Optimization is a parallelizable method for gradient estimation.
problem Gradient estimation for differentiable objectives in parallel environments.
method Variational Optimization, Natural Evolution Strategies, Gaussian Perturbation, Directional Derivatives.
result Directional Derivatives are preferable to Variational Optimization for parallel Stochastic Gradient Descent.
Proves existence of multiple solutions to a multiphasic equation on manifolds.
problem Existence of multiple solutions to a multiphasic equation with a small volume constraint.
method Lusternik-Schnirelmann and infinite-dimensional Morse theories, combined with isoperimetric theory and transversality theorem.
result Lower bound for the number of solutions depending on topological invariants.
The abstract discusses the existence of complex structures on spheres and their implications.
problem The existence of complex structures on the six sphere and their implications.
method Analyzing the parallelism and H-space multiplication on the seven sphere associated with almost complex structures on the six sphere.
result The integrability condition of the almost complex structure on the six sphere does not imply the homotopy associativity of the multiplication on the seven sphere.
The paper proposes a parallelizable clustering method for multivariate data.
problem The standard model-based clustering method assumes the same number of clusters per margin, which is often unrealistic.
method Developed a finite mixture model per margin with different numbers of clusters, and used a game-inspired algorithm to cluster multivariate data.
result The proposed method shows good performance in various scenarios and real datasets.