Einstein's philosophy uses differential identities to derive GR field equations.
problem Constructing field theories in alternative geometries.
method Explains differential identities and their role in GR and PAP-geometry.
result Derived a more general differential identity in PAP-geometry.
New conformal invariants discovered in absolute parallelism geometry.
problem Investigating conformal changes in absolute parallelism geometry.
method Using the Weitzenböck and Levi-Civita connections of an absolute parallelism space.
result Found new conformal invariants.
A field theory is constructed in the context of parameterized absolute parallelism geometry. The theory is shown to be a pure gravity one. It is capable of describing the gravitational field and a material distribution in terms of the geometric structure of the geometry used (the parallelization vector fields). Three t…
A Lie group has a unique metric when viewed as a flat absolute parallelism.
problem Understanding different metrics on Lie groups and their curvature properties.
method Shifted perspective of viewing Lie groups as flat absolute parallelisms.
result Lie groups have a canonical metric when viewed in this new way.
The paper classifies almost complex manifolds with non-degenerate torsion and their coverings.
problem Classification of almost complex manifolds with non-degenerate torsion.
method Analysis of homogeneous and non-homogeneous manifolds, classification of Lie algebras, covering spaces.
result Classification of manifolds with solvable Lie algebra up to coverings.
In this paper we provide a \emph{global} investigation of the geometry of parallelizable manifolds (or absolute parallelism geometry) frequently used for application. We discuss the different linear connections and curvature tensors from a global point of view. We give an existence and uniqueness theorem for a remarkab…
The aim of the present paper is to construct and investigate a Finsler structure within the framework of a Generalized Absolute Parallelism space (GAP-space). The Finsler structure is obtained from the vector fields forming the parallelization of the GAP-space. The resulting space, which we refer to as a Finslerized Pa…
Develops a new method for constructing absolute parallelisms on CR structures.
problem Constructing absolute parallelisms for CR structures in arbitrary dimensions.
method Bigraded Tanaka prolongation procedure to construct canonical absolute parallelisms.
result Unique canonical absolute parallelism found for CR structures with maximal infinitesimal symmetry.
Parallel algorithm for conformal parameterization of 3D surfaces.
problem Computational difficulties with high-resolution 3D surface meshes.
method Partitioning surfaces into subdomains, parallel local parameterization, partial welding for boundary integration, solving Laplace equation.
result Significant improvement in computational time and accuracy compared to existing methods.
Determined structure equations for specific CR hypersurfaces.
problem Explicitly determine structure equations of CR hypersurfaces.
method Used recently constructed canonical Cartan connection.
result Explicitly determined structure equations of 5D Levi 2-nondegenerate CR hypersurfaces.
We show how to construct absolutely exotic smooth structures on compact 4-manifolds with boundary, including contractible manifolds. In particular, we prove that any compact smooth 4-manifold W with boundary that admits a relatively exotic structure contains a pair of codimension-zero submanifolds homotopy equivalent t…
PSP dynamically learns structured sparsity in DNNs for better parallelism.
problem Efficiently compressing large DNNs to match hardware capabilities.
method Parameterized Structured Pruning (PSP) that dynamically learns the shape of DNNs through structured sparsity.
result PSP maintains prediction performance while creating substantial structured sparsity.
Generalized Tanaka prolongation ensures convergence of formal embeddings of complex manifolds.
problem Ensuring convergence of formal embeddings of complex manifolds under weaker conditions.
method Formulated and proved generalized Tanaka prolongation for geometric structures.
result Convergence of formal embeddings holds under weaker semi-positive normal bundle conditions.
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.
Absolute parallelism geometry is frequently used for physical applications. It has two main defects, from the point of view of applications. The first is the identical vanishing of its curvature tensor. The second is that its autoparallel paths do not represent physical trajectories. The present work shows how these de…
In this paper, we study Absolute Parallelism (AP-) geometry on the tangent bundle TM of a manifold M. Accordingly, all geometric objects defined in this geometry are not only functions of the positional argument x, but also depend on the directional argument y. Moreover, many new geometric objects, which have n…
A submanifold of a Riemannian symmetric space is called parallel if its second fundamental form is a parallel section of the appropriate tensor bundle. We classify parallel submanifolds of the Grassmannian $\rmG^+_2(\R^{n+2})$ which parameterizes the oriented 2-planes of the Euclidean space Rn+2\,. Our main resul…
The paper studies the geometry of Wigner caustics and decomposes curves into parallel arcs.
problem Understanding the geometry and properties of Wigner caustics of curves.
method Decomposing curves into parallel arcs to analyze the Wigner caustic's smooth branches, inflexion points, and singularities.
result New insights into the number of smooth branches, rotation number, inflexion points, and cusp singularities of the Wigner caustic.
Study weak Frenet frame for non-smooth curves with finite curvature and torsion.
problem Defining weak binormal and normal for non-smooth curves with finite total curvature and torsion.
method Piecewise linear methods and density argument applied to polygonal curves.
result Weak binormal and normal are rectifiable curves agreeing with total absolute torsion and vector product of tangent indicatrix and weak binormal.
Develops an alternative approach to Tanaka's prolongation of geometric structures.
problem Generalizing Tanaka's prolongation of G-structures to a broader class of geometric structures.
method Alternative constructive approach based on quasi-gradations and filtered vector spaces.
result Constructs an alternative method for describing local invariants and automorphism groups of Tanaka structures.
Hybrid actor-critic learns in complex action spaces.
problem Learning in complex, structured action spaces.
method Parallel sub-actor networks and a critic network.
result Hybrid PPO outperforms previous methods in parameterized action spaces.
Paper proposes a novel method to improve matrix completion with median loss for large datasets.
problem Matrix completion with absolute deviation loss for large-scale data.
method Proposes a refinement step using pseudo data to improve inefficient estimators of median matrix completion.
result Turns inefficient estimators into a rate (near-)optimal matrix completion procedure.
Affine λ-equidistants of convex polygons with parallel opposite sides have applications to isoperimetric inequalities.
problem Reconstruction and area estimates for affine λ-equidistants of convex polygons with parallel opposite sides. method Using Wigner caustics and centre symmetry sets.
result Proving a discrete version of the improved isoperimetric inequality.
Study shows CR models' rigidity, proving Beloshapka's conjecture.
problem CR model rigidity and Beloshapka's conjecture.
method Elie Cartan's classical method and weight analysis of structure equations.
result CR automorphism Lie groups do not contain nonlinear maps.
Gradient descent memorizes many Gaussians efficiently.
problem Memorizing many Gaussians with minimal parameters.
method Gradient descent on a depth-two neural network.
result One step of gradient descent memorizes $Ω\left(\frac{dq}{\log^4(d)}
ight)$ Gaussians.
Optimized parallel algorithms for identifying strong ties in data.
problem Identifying strong ties in data with varying distances and community sizes.
method Design and analysis of sequential and parallel algorithms for partitioned local depths.
result Optimized algorithms achieve up to 19.4x speedup in parallel execution.
This paper develops GPCA for probability distributions using Otto-Wasserstein geometry.
problem Analyzing modes of variation in datasets of probability measures.
method Geodesic Principal Component Analysis (GPCA) on Wasserstein space with neural networks.
result Identification of geodesic curves that capture modes of variation in probability distributions.
We classify non-dilatonic NS-NS type II supergravity backgrounds admitting a consistent absolute parallelism. They are all given by parallelised Lie groups admitting scalar flat bi-invariant lorentzian metrics. There are seven different classes, some of them containing moduli. For each class we determine the amount of …
This work proposes an efficient autoregressive model for text generation.
problem The challenge of generating high-quality text with autoregressive models.
method Introduces a cascaded decoding approach using Markov transformers to achieve sub-linear parallel time generation.
result Shows competitive accuracy/speed tradeoff compared to existing methods on five machine translation datasets.
A cylindrical stretch line is a stretch line, in the sense of Thurston, whose horocyclic lamination is a weighted multicurve. In this paper, we show that two correctly parameterized cylindrical lines are parallel if and only if these lines converge towards the same point in Thurston's boundary of Teichmüller space.
Teacher algorithm helps DRL learn diverse environments efficiently.
problem Teach DRL to learn in various, unknown environments efficiently.
method Transformed into a bandit problem, learns to sample environments.
result ALP-GMM models learning progress, improving curriculum design.
In this paper we consider planar sections and visual contours of co-dimension one affine immersions. The main theorem says that the third order Taylor expansion of the difference between the visual contour and planar section functions is exactly the cubic form. We also consider parameterizations on two dimensional affi…
This work proposes a mathematical framework for loss landscapes and optimization in deep neural networks.
problem The effectiveness of gradient-based optimization in over-parameterized neural networks.
method A modern view and mathematical framework of loss landscapes and efficient optimization in over-parameterized machine learning models.
result Wide neural networks satisfy the PL∗ condition, explaining (S)GD convergence to a global minimum. Sharp bounds on neural network approximation rates and widths.
problem Estimating approximation rates, metric entropy, and n-widths of shallow neural networks.
method Introducing smoothly parameterized dictionaries and providing upper and lower bounds.
result Sharp bounds on approximation rates, metric entropy, and n-widths for neural networks with various activation functions.
Unitary RNNs tackle vanishing/exploding gradients in long-term dependencies.
problem Vanishing and exploding gradients in RNNs for long-term dependencies.
method Proposes a unitary weight matrix with eigenvalues of absolute value 1, parametrized by structured matrices.
result Achieves state-of-the-art results in tasks with very long-term dependencies.
We investigate a local reparameterizaton technique for greatly reducing the variance of stochastic gradients for variational Bayesian inference (SGVB) of a posterior over model parameters, while retaining parallelizability. This local reparameterization translates uncertainty about global parameters into local noise th…
Neural model with parameterized algorithms improves graph CO problem solving.
problem Solving NP-hard graph combinatorial optimization problems efficiently and accurately.
method Combining neural models and parameterized algorithms to identify and handle hard and easy parts of CO instances.
result Framework produces superior solution quality and out-of-distribution generalization.
Study on equilibrium points of dynamical systems with multiple integrals.
problem Understanding the equilibrium points of dynamical systems with multiple independent first integrals.
method Analyzes the equilibrium locus as a smooth manifold and fiber bundle with a natural connection.
result Parallel transport exists for the connection and can measure eigenvalue variations.
Gradient descent solves non-convex neural networks with random initialization.
problem Gradient descent can solve non-convex neural networks with random initialization.
method Gradient descent, over-parameterized neural networks, random initialization, strong convexity-like property.
result Gradient descent converges to a globally optimal solution at a linear rate.
Recently, it has been shown that Absolute Parallelism (AP) geometry admits paths that are naturally quantized. These paths have been used to describe the motion of spinning particles in a background gravitational field. In case of a weak static gravitational field limits, the paths are applied successfully to interpret…
Classifies GL(2,R)-invariant subvarieties with zero Lyapunov exponents.
problem Classifying GL(2,R)-invariant subvarieties with specific properties.
method Classification based on homological dimensions and Lyapunov exponents.
result Explicit exceptions list for GL(2,R)-invariant subvarieties with zero Lyapunov exponents.
The paper addresses nonconvex penalized LAD estimation in partial linear models using DNNs.
problem Challenges in nonconvex penalized LAD estimation with DNNs in partial linear models.
method Parameterizes nonparametric term with DNNs, formulates penalized LAD problem, introduces proximal subgradient method.
result Establishes consistency, convergence rate, and asymptotic normality of the estimator.
We reduce to various absolute parallelisms, namely to certain {e}-structures on manifolds of dimensions 7, 6, 5, the biholomorphic equivalence problem or the intrinsic CR equivalence problem for generic submanifolds M^5 in C^4 of CR dimension 1 and of codimension 3 that are maximally minimal and are geometry-preserving…
In this paper, we consider real hypersurfaces M in C3 (or more generally, 5-dimensional CR manifolds of hypersurface type) at uniformly Levi degenerate points, i.e. Levi degenerate points such that the rank of the Levi form is constant in a neighborhood. We also require the hypersurface to satisfy a certain s…
A geometric structure (FAP-structure), having both absolute parallelism and Finsler properties, is constructed. The building blocks of this structures are assumed to be functions of position and direction. A non-linear connection emerges naturally and is defined in terms of the building blocks of the structure. Two lin…
Develops new approach to recover CR structures from their Levi foliations.
problem Recovering CR structures from their Levi foliations for nonregular symbols.
method Reduction to dynamical Legendrian contact structure on leaf space.
result New geometric interpretation of CR prolongation conditions.
The least absolute shrinkage and selection operator (lasso) and ridge regression produce usually different estimates although input, loss function and parameterization of the penalty are identical. In this paper we look for ridge and lasso models with identical solution set. It turns out, that the lasso model with shri…
We establish a relationship between Heegaard Floer homology and the fractional Dehn twist coefficient of surface automorphisms. Specifically, we show that the rank of the Heegaard Floer homology of a 3-manifold bounds the absolute value of the fractional Dehn twist coefficient of the monodromy of any of its open book d…