This study examines the space of initial values strictly satisfying the dominant energy condition.
problem The dominant energy condition on initial value pairs in spacetime.
method Introduced an index difference for initial value pairs and compared it to Riemannian metrics.
result The space of initial values has non-trivial homotopy groups.
Recent results using inverse scattering techniques interpret every solution φ(x,y) of the sine-Gordon equation as a non-linear superposition of solutions along the axes x=0 and y=0. Here we provide a geometric method of integration, as well as a geometric interpretation. Specifically, every weakly regular surface…
In this paper we study a model of random knots obtained by fixing a space curve in n-dimensional Euclidean space with n>3, and orthogonally projecting the space curve on to random 3 dimensional subspaces. By varying the space curve we obtain different models of random parametrized knots, and we will study how the…
New method for initializing low-rank neural networks improves performance.
problem Training low-rank neural networks efficiently and accurately.
method Inspired by function approximation, proposes a novel low-rank initialization framework.
result Demonstrates significant gap between spectral and low-rank initialization approaches.
Let us fix two different radial eigenfunctions of a hyperbolic Laplacian and assume that both of them have the same value at the origin. Both eigenvalues can be complex numbers. The main goal of this paper is to estimate the lower bound for the interval (0,T], where these two eigenfunctions must assume different values…
A new geometric cocycle measures mass of hyperbolic manifolds.
problem Measuring mass of hyperbolic manifolds.
method Constructing a cocycle mapping ALH metrics to a tractor-valued form field.
result Invariant c(h) can be integrated over a sphere boundary. PBVFs generalize across policies using learned value functions.
problem RL algorithms forget information about old policies when updating value functions to track the learned policy.
method Introduce Parameter-Based Value Functions (PBVFs) that include policy parameters in their inputs, enabling them to generalize across different policies.
result PBVFs enable zero-shot learning of new policies that outperform any policy seen during training.
Solves initial value problem for harmonic maps on specific manifolds.
problem Initial value problem for harmonic maps on cohomogeneity one manifolds.
method Setup and solve the initial value problem using equivariant harmonic maps and regular-singular systems.
result Local existence of harmonic maps in a neighborhood of singular orbits.
Finite singular times for symmetric network curvature flow.
problem Formation of singularities in network curvature flow.
method Curvature flow of networks with symmetric initial data and two triple junctions.
result The set of singular times is finite.
We show stability of pairs of Ricci flat metrics and parallel spinor fields with respect to the spinor flow, i.e. we show that the spinor flow with initial conditions near such pairs converges to a critical point with exponential speed. Moreover, we show stability of certain volume constrained critical points of the sp…
The paper generalizes a mean value theorem for solutions of the ultrahyperbolic equation.
problem Establishing a mean value property for solutions of the ultrahyperbolic equation.
method Using conformal maps of the pseudo-Euclidean space of signature 2+2, the paper extends Asgeirsson's theorem to a more general class of pairs of curves.
result The mean value property is proven for non-degenerate conjugate conics, including conjugate circles, hyperbolae, parabolae, and line-empty pairs.
Solves initial boundary value problem for vacuum Einstein equations and proves geometric uniqueness.
problem Initial boundary value problem for vacuum Einstein equations.
method Formulated IBVP, solved simultaneously in local harmonic coordinates, constructed unique maximal globally hyperbolic solution.
result Vacuum spacetimes satisfying fixed initial-boundary conditions and corner conditions are geometrically unique near the initial surface.
Fenrir uses probabilistic numerics to simplify solving initial value problems.
problem Solving initial value problems in ordinary differential equations.
method Probabilistic numerics and Gauss--Markov regression.
result The method simplifies parameter estimation in ODEs, making it easier and more robust.
We consider the gradient flow of the Yang-Mills-Higgs functional of twist Higgs pairs on a Hermitian vector bundle (E,H0) over a Riemann surface X. It is already known the gradient flow with initial data (A0,φ0) converges to a critical point (A∞,φ∞) of this functional. Using a modified Chern-Wei…
Improved neural network training by coupled initialization reduces neuron count.
problem Training neural networks efficiently with fewer neurons.
method Coupled initialization of weights into pairs of identical Gaussian vectors.
result Significantly reduced number of neurons required for network convergence.
We construct differential equivariant K-theory of representable smooth orbifolds as a ring valued functor with the usual properties of a differential extension of a cohomology theory. For proper submersions (with smooth fibres) we construct a push-forward map in differential equivariant K-theory. Finally, we construct …
Geometrically interpolates rigid body motions with initial and terminal twists.
problem Finding spatial trajectories between prescribed initial and terminal poses.
method Derives solutions for k-IV-TIP and k-BV-TIP for k=1,...,4.
result Automatic cubic interpolation identical to minimum acceleration curve when twists are zero.
Imputation-free method learns tabular data with missing values using transformer.
problem Machine learning on tabular data with missing values often leads to unreliable outcomes due to synthetic imputation.
method Incremental attention learning (IFIAL) using transformer with attention masks.
result IFIAL outperforms state-of-the-art methods in 17 diverse tabular data sets.
Paper investigates multimodal contrastive learning and incorporates unpaired data.
problem Improving feature learning ability of multimodal models under noisy data.
method Initiates investigation of nonlinear loss functions for multimodal contrastive learning, analyzes performance, proposes new loss incorporating unpaired data.
result MMCL can outperform unimodal contrastive learning and robustly handle noisy data.
New sampling methods improve Shapley values for explaining machine learning predictions.
problem Computational limitations in calculating Shapley values for complex models.
method Asymptotic normality results and paired-sampling approximations (KernelSHAP and PermutationSHAP).
result Paired-sampling PermutationSHAP provides exact results for interactions of maximal order two and has the additive recovery property.
This article, written to appear as a chapter in "The Springer Handbook of Spacetime", is a review of the initial value problem for Einstein's gravitational field theory in general relativity. Designed to be accessible to graduate students who have taken a first course in general relativity, the article first discusses …
Study TMF-valued TQFT for closed 3-manifolds using torsion linking pairings.
problem Explicit description of TMF-module state space for closed 3-manifolds.
method Construct canonical invariants and tokenization from torsion linking pairings.
result Explicit model for TMF-module state space in terms of rank-one TMF-module.
Constructs unique bases for CY varieties over valued fields.
problem Finding unique bases for CY varieties over valued fields.
method Uses techniques from higher rank degenerations in K-stability.
result Induces canonical functions on skeletons and agrees with tropicalizations of theta functions.
Machine learning predicts flight connections for airline crew scheduling.
problem Predicting the next connecting flight for airline crews.
method Adapted neural network for multiclass classification from historical data.
result High accuracy (99.7%) in flight connection prediction.
In distributed function computation, each node has an initial value and the goal is to compute a function of these values in a distributed manner. In this paper, we propose a novel token-based approach to compute a wide class of target functions to which we refer as "Token-based function Computation with Memory" (TCM) …
New boundary conditions improve Hamiltonian analysis in GR.
problem Improving Hamiltonian analysis in GR with IBVP.
method Presented and analyzed new boundary conditions.
result New boundary conditions lead to better Hamiltonian analysis.
Study examines how twisting graphene nanoribbons affects their thermal conductivity.
problem Understanding how twisting affects thermal conductivity in graphene nanoribbons.
method Calculated geometric parameters of TGNRs, including twist and writhe, and used molecular dynamics simulations.
result Twisted graphene nanoribbons require at least two parameters to accurately describe their thermal conductivity.
In this paper, we develop an alternating direction method of multipliers (ADMM) for deep neural networks training with sigmoid-type activation functions (called \textit{sigmoid-ADMM pair}), mainly motivated by the gradient-free nature of ADMM in avoiding the saturation of sigmoid-type activations and the advantages of …
The Bellman error is a poor proxy for value function accuracy, even with all state-action pairs.
problem The Bellman error is a poor proxy for the accuracy of the value function.
method Study of the Bellman equation as a surrogate objective for value prediction accuracy.
result The magnitude of the Bellman error is only weakly related to the distance to the true value function, even with all state-action pairs.
Defines height pairing for differential forms on Riemann surface degenerations.
problem Calculating heights for differential forms on degenerating Riemann surfaces.
method Defines Archimedean height pairing, uses Dai-Yoshikawa asymptotics, extends Filip-Tosatti construction.
result Relates new pairing to current-valued pairing, extends geometric settings.
In this paper, we consider the gradient flow of the Yang-Mills-Higgs functional for Higgs pairs on a Hermitian vector bundle (E,H0) over a compact Kähler manifold (M,ω). We study the asymptotic behavior of the Yang-Mills-Higgs flow for Higgs pairs at infinity, and show that the limiting Higgs sheaf is isomorph…
Paper enhances RL policies using trust region optimization for offline data.
problem Limited data in offline RL settings.
method Trust region optimization for value enhancement.
result Enhanced policy values with faster convergence.
Let Sg denote the closed orientable surface of genus g. We construct exponentially many mapping class group orbits of pairs of simple closed curves which fill Sg and intersect minimally, by showing that such orbits are in correspondence with the solutions of a certain permutation equation in the symmetric g…
New foliations constructed from contact pairs, revealing flexible taut foliations.
problem Characterizing taut foliations in 3D.
method Construction of codimension-one foliations from pairs of contact structures in 3D.
result Foliations constructed are taut, providing new insights into the L-space conjecture.
Blockchain-based exchanges adopt based on token pair volatility and personal use.
problem Token value loss and arbitrage issues in decentralized exchanges.
method Investigation of Automated Market Makers (AMMs) using transaction-level data.
result AMMs are adopted for high personal use or highly correlated token price movements.
Study examines persistence diagrams in machine learning, proposing permutation tests.
problem Understanding the power and limitations of persistence diagrams in machine learning.
method Carried out experiments on graph and shape data, proposed permutation tests for persistence diagrams.
result Persistence pairing shows significant improvement in various tasks, but the most critical values are most discriminative.
Develops moduli theory for Calabi-Yau pairs, constructing a projective space.
problem Constructing a moduli space for Calabi-Yau pairs at the Calabi-Yau wall.
method Develops moduli theory, proving S-completeness and Θ-reductivity, constructing projective moduli space. result Constructs a projective moduli space for degenerate P2 pairs. Explains neural network properties with a simple operation.
problem Understanding neural network performance and generalization.
method Analyze a single operation: expand-and-sparsify.
result Clarifies phenomena like Lottery Ticket Hypothesis and overparameterized models.
GP-ND avoids obstacles in trajectory planning using Gaussian Process regression.
problem Avoiding obstacles in trajectory planning for real-world systems.
method GP-ND models negative data pairs using Gaussian distributions and maximizes their KL divergence from the GP to avoid them.
result GP-ND outperforms traditional GP learning in obstacle-aware trajectory planning.
New algorithm calibrates local volatility from option prices using deep neural networks.
problem Calibrating local volatility from market option prices with reduced interpolation and reprice errors.
method Deep self-consistent learning using neural networks to approximate both option prices and local volatility.
result Improved performance in terms of reduced interpolation and reprice errors compared to existing methods.
Investigates parallel spinors on Lorentzian four-manifolds using differential geometry.
problem Characterizing and classifying Lorentzian four-manifolds with parallel spinors.
method Formulated parallel spinor flow equations and used parabolic pairs theory.
result Characterized all parallel Cauchy pairs on simply connected Cauchy surfaces and classified compact three-manifolds.
A novel graphical matching approach improves pairs trading by reducing portfolio variance and risk-adjusted returns.
problem Common pairs trading methods lead to high portfolio variance and low risk-adjusted returns due to focusing on highly cointegrated assets.
method Model all assets and their cointegration levels with a weighted graph. Select pairs as a maximum weighted matching to ensure no shared assets and lower portfolio variance.
result The matching-based strategy shows a significant improvement in risk-adjusted performance, with a gross Sharpe ratio of 1.23.
We consider pairs of a non-empty compact connected and locally connected Hausdorff space and a real-valued continuous function. Our aim is to measure the difference between this kind of the pairs. In this notes we introduce new pseudodistances between pairs associated with reparametrization invariant seminorms. We fini…
Proves spacetime positive mass theorem for spin initial data sets with arbitrary ends.
problem Proving the spacetime positive mass theorem for specific spacetime configurations.
method Solving a mixed boundary value problem for the Dirac-Witten operator with a Callias potential.
result Established spacetime positive mass theorem for asymptotically flat spin initial data sets with arbitrary ends.
In this paper we study the local regularity of closed surfaces immersed in a Riemannian 3-manifold flowing by Willmore flow. We establish a pair of concentration-compactness alternatives for the flow, giving a lower bound on the maximal time of existence of the flow proportional to the concentration of the curvature an…
Formula connects complex forms on deforming spaces.
problem Establishing a formula for complex forms on deforming spaces.
method Holomorphic family of pairs and correspondence formula.
result Formula connects Dolbeault complexes of deforming spaces.
It is well known that the initialization of weights in deep neural networks can have a dramatic impact on learning speed. For example, ensuring the mean squared singular value of a network's input-output Jacobian is O(1) is essential for avoiding the exponential vanishing or explosion of gradients. The stronger condi…
Firstly, we wish to motivate that Conley pairs, realized via Salamon's definition [17], are rather useful building blocks in geometry: Initially we met Conley pairs in an attempt to construct Morse filtrations of free loop spaces [21]. From this fell off quite naturally, firstly, an alternative proof [20] of the cell a…