Carter tensor analysis aids wave equation on Kerr-Newman spacetime.
problem Analyzing perturbations of Kerr-Newman spacetime using wave equation.
method Physical-space analysis adapted to Kerr-Newman spacetime, leveraging Carter operator commutation.
result Carter operator commutes with wave equation on Kerr-Newman spacetime, enabling wave equation analysis.
Study axisymmetric waves on extremal Kerr spacetime using physical-space estimates.
problem Obtain integrated local energy decay estimates for axisymmetric waves on extremal Kerr backgrounds.
method Use physical-space analysis and a method introduced by Stogin, simplifying Aretakis' derivation.
result Extend Morawetz estimates to extremal Kerr spacetime using purely classical currents.
Global existence and decay for quasilinear wave equations on various spacetimes, including Kerr black holes.
problem Global existence and decay for quasilinear wave equations on asymptotically flat spacetimes.
method Dyadically localised nature and direct use of a blackbox linear inhomogeneous energy estimate on exactly stationary metrics.
result Global existence and decay for small-data solutions to quasilinear wave equations on a wide variety of spacetime backgrounds, including Kerr black holes.
This paper proposes grid cells encode position via a conformal isometric embedding of 2D physical space.
problem Hexagonal grid firing patterns in grid cells.
method Learning a distance-preserving position embedding in neural space using a recurrent neural network.
result The conformal isometric embedding of 2D physical space into neural space explains hexagonal grid firing patterns.
New method finds precise late-time behavior of wave equations.
problem Analyzing late-time behavior of wave equations with inverse-square potentials.
method Physical-space-based method for deriving late-time asymptotics.
result Sharp, uniform decay estimates in time for asymptotic late-time tails.
Physics-informed neural operator learns from coarse to fine discretized data.
problem Lack of high-fidelity training data and uneven grid resolution.
method Physics-informed multi-resolution neural operator framework.
result Learn from arbitrarily discretized input functions using latent embedding and finite difference solver.
The paper proves boundedness and decay of Teukolsky equations on Kerr backgrounds.
problem Analyzing boundedness and decay of Teukolsky equations on Kerr backgrounds.
method Adapting techniques from scalar waves, uniform-in-frequency estimates for Teukolsky PDEs were obtained.
result Solutions of Teukolsky equation on subextremal Kerr backgrounds remain bounded and decay in time.
New insights into data geometry reveal manifold structure in grid-cell activity.
problem Understanding the roles of different dimensions in data geometry.
method Generalised Hanson-Wright inequality and random function model analysis.
result Persistence diagrams reveal latent homology and manifold structure.
HFNO enhances interpretability of turbulent flows through parallel wavenumber bin processing.
problem Opaque inner workings of Fourier Neural Operators (FNOs) hinder physical interpretability.
method Introduces HFNO, a novel FNO-based architecture that processes wavenumber bins in parallel, enhancing interpretability.
result HFNO decomposes turbulent flows across various scales, enabling increased interpretability and multiscale modeling.
We develop a cross-sectional research design to identify causal effects in the presence of unobservable heterogeneity without instruments. When units are dense in physical space, it may be sufficient to regress the "spatial first differences" (SFD) of the outcome on the treatment and omit all covariates. The identifyin…
New method reduces high-dimensional data to key features.
problem Challenges of high-dimensional data analysis and interpretability.
method Randomized search to produce subspaces, ensemble of models for variable selection.
result Outperforms existing methods in prediction and variable selection.
The fundamental group and rational cohomology of the configuration spaces of the Skyrme and Faddeev-Hopf models are computed. Physical space is taken to be a compact oriented 3-manifold, either with or without a marked point representing an end at infinity. For the Skyrme model, the codomain is any Lie group, while for…
New methods estimate multivariate shortfall risk more efficiently.
problem Estimating multivariate shortfall risk is computationally challenging.
method Combines Fourier inversion and RQMC sampling in frequency domain.
result Fourier RQMC methods outperform existing benchmarks.
A collaborative convex framework for factoring a data matrix X into a non-negative product AS, with a sparse coefficient matrix S, is proposed. We restrict the columns of the dictionary matrix A to coincide with certain columns of the data matrix X, thereby guaranteeing a physically meaningful dictionary and …
We present a variational renormalization group (RG) approach using a deep generative model based on normalizing flows. The model performs hierarchical change-of-variables transformations from the physical space to a latent space with reduced mutual information. Conversely, the neural net directly maps independent Gauss…
Scattering theory developed for linearised gravity near Schwarzschild black hole.
problem Linear stability of Schwarzschild spacetime and scattering of gravitational waves.
method Physical-space Chandrasekhar transformation and Teukolsky-Starobinsky correspondence.
result Construction of scattering theory for spin 2 Teukolsky equations.
The paper explores Kaluza-Klein theories without assuming a fibration structure.
problem Exploring Kaluza-Klein theories without assuming a fibration structure.
method Variational formulations of gauge theories and Einstein--Yang-Mills equations.
result Classical solutions allow the construction of a manifold X of dimension 4 as physical space-time, leading to solutions of the Einstein--Yang-Mills systems. 4-dim intrinsic (material) Riemannian metric G of the material 4-D space-time continuum P is utilized as the characteristic of the aging processes developing in the material. Manifested through variation of basic material characteristics such as density, moduli of elasticity, yield stress, strength, and toughness.,…
The paper tackles drift identification in Lévy α-stable stochastic systems, proposing a Fourier space approach.
problem Estimating the drift field of a stochastic differential equation driven by Lévy α-stable noise.
method Fourier space approach, parameterizing the drift field using Fourier series, minimizing a loss function with gradients computed via the adjoint method.
result The method is capable of learning drift fields in qualitative and/or quantitative agreement with ground truth fields.
Proves stability of Schwarzschild black holes without symmetry assumptions.
problem Stability of Schwarzschild black holes under general conditions.
method Teleologically normalised double null gauges, analysis of linear stability, and control of non-linearities.
result Proves non-linear asymptotic stability of Schwarzschild family as solutions to Einstein vacuum equations.
We prove boundedness and polynomial decay statements for solutions to the spin ±1 Teukolsky-type equation projected to the ℓ=1 spherical harmonic on Reissner-Nordström spacetime. The equation is verified by a gauge-invariant quantity which we identify and which involves the electromagnetic and curvature tensor…
We prove in this paper the linear stability of the celebrated Schwarzschild family of black holes in general relativity: Solutions to the linearisation of the Einstein vacuum equations around a Schwarzschild metric arising from regular initial data remain globally bounded on the black hole exterior and in fact decay to…
Second part of series studying charged scalar fields on Reissner--Nordström spacetimes.
problem Analyzing late-time behavior and stability of charged scalar fields on black hole backgrounds.
method Purely physical-space based methods, energy estimates, inverse-power laws.
result First pointwise decay estimates for charged scalar fields on black hole backgrounds.
A geometric string solution has background fields in overlapping coordinate patches related by diffeomorphisms and gauge transformations, while for a non-geometric background this is generalised to allow transition functions involving duality transformations. Non-geometric string backgrounds arise from T-duals and mirr…
In the historical literature there has been an extended discussion on the question, whether the report of Sartorius von Waltershausen about C. F. Gauss checking the largest triangle of the geodetical measurement campaign in the kingdom of Hannover as a kind of ``test'' for the Euclididean nature of physical space can b…
We focus in this paper on high-dimensional regression problems where each regressor can be associated to a location in a physical space, or more generally a generic geometric space. Such problems often employ sparse priors, which promote models using a small subset of regressors. To increase statistical power, the so-c…
Motivated by the sigma model limit of multicomponent Ginzburg-Landau theory, a version of the Faddeev-Skyrme model is considered in which the scalar field is coupled dynamically to a one-form field called the supercurrent. This coupled model is investigated in the general setting where physical space is an oriented Rie…
Hexagon grid patterns emerge from conformal isometry in grid cell neural networks.
problem Understanding the algebraic, geometric, and topological properties of grid cells.
method Investigating recurrent neural network models of grid cells, focusing on Lie group and Lie algebra representations, conformal isometry, and hexagon periodic patterns.
result Conformal isometry leads to hexagon periodic patterns in grid cell responses and accurate path integration.
The paper studies how grid cell patterns emerge in neural networks.
problem Understanding how grid cells in the brain form hexagonal firing patterns.
method Training recurrent neural networks with conformal normalization of velocity inputs.
result Conformal normalization is crucial for the emergence of hexagonal grid patterns in neural networks.
A new method optimizes Fourier pricing for multi-asset options using adaptive quadrature.
problem Efficiently pricing multi-asset options in Lévy models.
method Optimized damping parameters and hierarchical adaptive quadrature.
result Significant speed-up in computational time for up to six dimensions.
New method reconstructs Black-Scholes option prices from current profiles.
problem Reconstructing Black-Scholes prices from current profiles, dealing with ill-posedness.
method Price-dimensional reduction using Legendre polynomials, Tikhonov regularization.
result Reconstructs Black-Scholes prices from noisy initial data, stabilizing the solution.
We develop a definitive physical-space scattering theory for the scalar wave equation on Kerr exterior backgrounds in the general subextremal case |a|<M. In particular, we prove results corresponding to "existence and uniqueness of scattering states" and "asymptotic completeness" and we show moreover that the resulting…
This paper contains the first two parts (I-II) of a three-part series concerning the scalar wave equation \Box_gψ = 0 on a fixed Kerr background. We here restrict to two cases: (II1) |a| \ll M, general ψ or (II2) |a| < M, ψ axisymmetric. In either case, we prove a version of 'integrated local energy decay', specificall…
Global existence and boundedness proved for quasilinear wave equations on Kerr black holes.
problem Global existence and boundedness for quasilinear wave equations on Kerr black holes.
method Combines linear inhomogeneous estimates on Kerr backgrounds and tailored physical space currents.
result Global existence, boundedness and decay for small data solutions to quasilinear wave equations on Kerr black holes.
Efficient private matrix analysis algorithms for recent variants.
problem Private analysis of recent matrix updates.
method Identifying sufficient conditions on positive semidefinite matrices.
result First efficient differentially private algorithms for various matrix analysis tasks.
This paper introduces compositional data analysis for financial ratios, improving industry-level analysis.
problem Statistical issues with standard financial ratios at industry level.
method Compositional data analysis techniques for financial ratios.
result Improved analysis of financial ratios using compositional data methods.
In this dissertation, the main goal is visualisation of financial time series. We expect that visualisation of financial time series will be a useful auxiliary for technical analysis. Firstly, we review the technical analysis methods and test our trading rules, which are built by the essential concepts of technical ana…
Paper combines geometry and time-series analysis for spatiotemporal data.
problem Multivariate time-series data from multiple sensors.
method Combines manifold learning, Riemannian geometry, and spectral analysis.
result Proposes Riemannian multi-resolution analysis (RMRA) for dynamic mode extraction.
In this paper the exact linear relation between the leading eigenvectors of the modularity matrix and the singular vectors of an uncentered data matrix is developed. Based on this analysis the concept of a modularity component is defined, and its properties are developed. It is shown that modularity component analysis …
This paper investigates to identify the requirement and the development of machine learning-based mobile big data analysis through discussing the insights of challenges in the mobile big data (MBD). Furthermore, it reviews the state-of-the-art applications of data analysis in the area of MBD. Firstly, we introduce the …
Interactive DR framework for comparing datasets.
problem Limited flexibility in existing DR methods for comparative analysis.
method Unified linear comparative analysis (ULCA) with interactive optimization and visualization.
result ULCA and optimization algorithm improve comparative analysis efficiency and flexibility.
Combines topological and geometric approaches to data analysis.
problem Understanding when and how geometric objects intersect.
method Connects topological and geometric concepts of curvature.
result Reconceptualizes curvature and links it to hyperconvexity.
Proposes a multivariate regression model for better analysis of multiple datasets.
problem Insufficient performance of single-dataset analysis in integrative studies.
method Sparse estimation for variable and group selection, alternating direction method of multipliers algorithm.
result Demonstrated improved performance through simulations and real data analysis.
New method uses topological data analysis to study stock market crashes.
problem Characterizing and predicting stock market crashes.
method Topological data analysis, persistence landscape, dynamic time series analysis.
result Demonstrates effectiveness of new method for Flash Crash characterization and prediction.
Analyzes stock trends and e-commerce user behavior using Twitter data.
problem Understanding the relationship between stock prices, stock news, and e-commerce user behavior.
method Cross-domain analysis using Hadoop, Hive, and Tableau on three datasets.
result Identified correlations between stock sentiment, stock trends, and e-commerce user behavior.
Study analyzes Disney stock market performance using machine learning.
problem Forecasting stock market performance of Disney.
method Exploratory data analysis, feature engineering, model selection (linear regression).
result Linear regression model performed best.
This study analyzes data science vocabulary changes over 13 years.
problem Understanding evolution of data science terms over time.
method Exploratory Data Analysis, Latent Semantic Analysis, Latent Dirichlet Analysis, N-grams Analysis.
result Identified new vocabulary and its incorporation into scientific literature.
FinSphere improves stock analysis quality with AI and expert-curated data.
problem Lack of objective evaluation metrics and depth in stock analysis by FinLLMs.
method Developed AnalyScore, curated Stocksis dataset, and FinSphere AI agent.
result FinSphere outperforms general and domain-specific LLMs in generating high-quality stock analysis reports.