Quantizes area of surfaces in loop quantum gravity.
problem Quantizing the area of surfaces in loop quantum gravity.
method Using hyperlink and holonomy operators, quantize area of surfaces.
result Area operator can be computed from link-surface diagrams.
Survey on strong convergence in random matrices and its applications.
problem Understanding convergence of random matrices to operators.
method Analysis of operator norms of noncommutative polynomials.
result New insights and applications in random graphs, geometry, and operator algebras.
4-manifolds with nonnegative sectional curvature are area-extremal.
problem Finding extremal properties of 4-manifolds with curvature constraints.
method Analyzing sections in the kernel of a twisted Dirac operator and using the Finsler--Thorpe trick.
result Large classes of compact 4-manifolds are area-extremal.
Maps on certain manifolds decrease area and scalar curvature is bounded.
problem Understanding maps on manifolds and their impact on scalar curvature.
method Simple deformation of the Dirac operator.
result If scalar curvature is nonnegative and maps decrease area, scalar curvature must be negative.
Geometric phases describe how in a continuous-time dynamical system the displacement of a variable (called phase variable) can be related to other variables (shape variables) undergoing a cyclic motion, according to an area rule. The aim of this paper is to show that geometric phases can exist also for discrete-time sy…
LUTNet optimizes FPGA neural network accelerators by leveraging LUTs for inference, achieving significant area savings.
problem Redundancy in deep neural networks and inefficient use of FPGA resources.
method End-to-end hardware-software framework using LUTs to implement any K-input Boolean operation for inference.
result Significant area savings and comparable accuracy compared to state-of-the-art binarized neural networks.
The existence of kinematic formulas for area measures with respect to any connected, closed subgroup of the orthogonal group acting transitively on the unit sphere is established. In particular, the kinematic operator for area measures is shown to have the structure of a co-product. In the case of the unitary group the…
New operators help focus on specific areas in complex math problems.
problem Concentration in complex mathematical structures.
method Construct conjugate-linear perturbations of twisted spinc Dirac operators using the conjugate-linear Hodge star operator.
result These perturbations satisfy the concentration principle.
Paper studies eigenvalue bounds for complex curves on Kähler surfaces.
problem Investigates eigenvalue bounds for complex curves on Kähler surfaces.
method Analyzes second variation of a conformally invariant Willmore-type functional to derive bounds.
result Derives lower bound Λ1≥2Ric for Kähler surfaces, with equality for low genus curves. Smooth cones can be approximated by smooth hypersurfaces.
problem Approximating cones with smooth surfaces.
method Hyperbolic unfoldings and Jacobi field operator potential theory.
result Every area minimizing cone can be approximated by smooth hypersurfaces.
New theory for area of Legendrian surfaces, proving smoothness and variational results.
problem Understanding the area of Legendrian surfaces under constraints.
method Introducing PHSLVs, proving sequential compactness, regularity, and variational results.
result Generalized regularity theory for Legendrian surfaces, achieving variational minima.
A new metric estimates classifier accuracy using only training data.
problem Assessing classifier accuracy without cross-validation.
method Bayesian Area Under the ROC Curve (CBAUC) metric for linear classifiers.
result The CBAUC is faster and more accurate than conventional AUC estimators.
The study finds a continuous map achieving minmax area under Legendrian constraints.
problem Finding minmax areas under Legendrian constraints in 5D Sasakian manifolds.
method Continuous conformal Legendrian map with bounded multiplicity satisfying a weak Hamiltonian Minimal Equation.
result Continuous map achieving minmax area with bounded multiplicity.
LUTNet optimizes FPGA for neural network inference, achieving high efficiency.
problem Redundancy in neural networks leads to inefficient hardware implementations.
method Exploits LUTs' flexibility for efficient neural network inference on FPGAs.
result Significant area savings with comparable accuracy compared to state-of-the-art binarized neural networks.
We describe the asymptotic behavior of minimal area submanifolds in product spacetimes of an asymptotically hyperbolic space times a compact internal manifold. In particular, we find that unlike the case of a minimal area submanifold just in an asymptotically hyperbolic space, the internal part of the boundary submanif…
This expository paper, based on a Current Events Bulletin talk at the January, 2016 Joint Meetings, introduces the concept of Lyapunov exponents and discusses the role they play in three areas: smooth ergodic theory, Teichmüller theory, and the spectral theory of one-frequency Schrödinger operators. The inspiration for…
Plateau's problem is to show the existence of an area minimizing surface with a given boundary, a problem posed by Lagrange in 1760. Experiments conducted by Plateau showed that an area minimizing surface can be obtained in the form of a film of oil stretched on a wire frame, and the problem came to be called Plateau's…
The paper studies energy functionals for Lagrangian tori in complex projective space.
problem Investigating energy functionals for Lagrangian tori in complex projective space.
method Introducing an energy functional based on the potential of associated Schrödinger operators and studying its behavior on specific families of tori.
result Proposes that the minimum of the energy functional is achieved by the Clifford torus.
Minimal Gaussian surface area sets must be round cylinders if they are convex.
problem Finding sets with minimal Gaussian surface area under symmetry constraints.
method Colding-Minicozzi theory for Gaussian minimal surfaces and randomly chosen degree 2 polynomial.
result Convex sets with minimal Gaussian surface area are round cylinders.
Koopman operator theory simplifies complex systems analysis.
problem Analyzing nonlinear dynamical systems and complex networks.
method Estimating Koopman operator from data to reveal system properties.
result Koopman operators provide insights into system characteristics.
We prove an isoperimetric inequality for the second non-zero eigenvalue of the Laplace-Beltrami operator on the real projective plane. For a metric of the unit area this eigenvalue is not greater than 20π. This value is attained in the limit by a sequence of metrics of area one on the projective plane. The limiting met…
Whyte used the index theory of Dirac operators and Block-Weiberger uniformly finite homology to show that certain infinite connected sums do not carry a metric with nonnegative scalar curvature in their bounded geometry class. His proof uses a coarse version of the A^-class to obstruct such metrics. In this note…
We analyze the level sets of the norm of the Witten spinor in an asymptotically flat Riemannian spin manifold of positive scalar curvature. Level sets of small area are constructed. We prove curvature estimates which quantify that, if the total mass becomes small, the manifold becomes flat with the exception of a set o…
Foundation models outperform supervised methods in time series forecasting across various operational regimes.
problem Lack of domain-specific training and ongoing maintenance in supervised learning for time series forecasting.
method Evaluation of foundation models against standard supervised approaches across four operational regimes: periodic, physically constrained, stochastic, and demand forecasting.
result Foundation models are optimal for cold-start or long-tail scenarios and perform well in domains with transferable periodic structures.
Reanalysis of bioactivity prediction models suggests SVM performance is competitive with deep learning.
problem Benchmarking and validation of machine learning models in drug discovery.
method Reanalysis of a large-scale comparison of machine learning models for bioactivity prediction, using numerical experiments to question ROC curve relevance and suggest precision-recall curve.
result Support vector machines show competitive performance with deep learning methods in bioactivity prediction.
Study proposes new methods to calculate probabilistic benchmarks in noisy data.
problem Identifying opportunities for improvement in comparable units with noisy data.
method 2-step methodology involving undersampling and relevance vector machine.
result Higher discrimination power achieved with macro-economic environment variables.
GIT-Net uses neural networks to approximate PDE operators efficiently.
problem Approximating PDE operators for complex geometries.
method Parametrizes adaptive generalized integral transforms with deep neural networks.
result GIT-Net outperforms existing neural network operators in multiple areas.
Machine learning predicts circulatory failure in ICU patients.
problem Limited ability of clinicians to recognize early signs of patient deterioration.
method Developed an early warning system using machine learning on ICU data.
result Predicts 90.0% of circulatory failure events with 81.8% identified more than two hours in advance.
EDAs with matrix transpose improve Bayesian structure learning performance.
problem Improving Bayesian structure learning performance.
method Introducing a matrix transpose mutation operator for EDAs in Bayesian structure learning.
result EDAs with transpose mutation give markedly better performance than conventional EDAs.
Estimates for polynomial operators using determinant majorization and subharmonics.
problem Bounding solutions of polynomial operators on Euclidean domains.
method Combines Alexandrov estimate and determinant majorization, using subharmonics and semiconvex approximation.
result Includes classical Alexandrov-Bakelman-Pucci estimate for linear operators.
Large filters improve performance but are costly; this work uses learned box filters and summed-area tables.
problem Improving performance in dense prediction tasks like human pose estimation with large filters.
method Adopted learnable box filters and summed-area tables to reduce computational cost and maintain performance.
result Demonstrated competitive performance on human pose estimation benchmarks.
Study on Dirac operator spectrum on hyperbolic surfaces with shrinking geodesics.
problem Spectrum of spin Dirac operator on hyperbolic surfaces with pinched geodesics.
method Trace formula for Dirac operator, Huber's theorem, small-time heat trace asymptotic expansion.
result Convergence of Selberg zeta function for degenerating hyperbolic surfaces.
The paper bounds eigenvalues of the Jacobi operator and derives rigidity results for CMC hypersurfaces.
problem Bounding the first eigenvalue of the Jacobi operator for CMC hypersurfaces.
method Geometric upper bounds for eigenvalues and rigidity results.
result New rigidity results for the area and length of CMC hypersurfaces.
Extends width estimates to family case using index theory.
problem Sharp width estimates for Riemannian bands with positive scalar curvature.
method Dirac operators and family index theory.
result Proves width estimate for fiber bundles with infinite A-hat area.
Automates neural network design for diverse tasks.
problem Designing neural networks for new, under-explored domains.
method Introduces XD-Operations and a weight-sharing scheme to transform standard backbones into search spaces of operations.
result Models using XD-Operations achieve lower error than baseline and expert-designed networks on diverse tasks.
GenUQ uses generative models to estimate uncertainty in operator learning.
problem Uncertainty quantification in stochastic operator models.
method Introduces a measure-theoretic approach with a generative hyper-network.
result Outperforms other UQ methods in various example problems.
New bounds on Bartnik mass for surfaces with non-negative first eigenvalue.
problem Bounding Bartnik mass for surfaces with spectral non-negativity condition.
method Proving upper bound on Bartnik mass using spectral non-negativity condition.
result Bounded above by √(|S²|_g/16π) under spectral non-negativity.
Random feature method approximates operators with theoretical guarantees and reduced computation.
problem Approximating operators between infinite dimensional Banach spaces using machine learning.
method Random feature operator learning method with theoretical guarantees and error bounds.
result The random feature method can achieve similar or better test errors than kernel-based methods and neural networks with significantly reduced training times.
Study on null-torsion holomorphic curves in 6-sphere, focusing on their second variation.
problem Characterize the second variation of area for null-torsion holomorphic curves in the round 6-sphere.
method Analyzing the spectrum of the Jacobi operator for compact null-torsion holomorphic curves.
result For g≤6, the multiplicity of the lowest eigenvalue λ1=−2 is exactly 4d. Study predicts antimicrobial resistance in ICU patients quickly.
problem Delayed AMR testing in ICU leads to suboptimal treatment.
method Developed predictive models using clinical and microbiological data.
result Machine learning models predict AMR with higher accuracy than naive model.
Introduces noncommutative geometry for modeling quantum spacetime.
problem Modeling quantum spacetime.
method Operator algebras, K-theory, spectral geometry, quantum groups, and deformation quantization.
result Framework for quantum spacetime.
This is the second in a series of papers where we estab- lish skin structural concepts and results for singular area minimizing hypersurfaces. Here we conformally unfold these spaces to complete Gromov hyperbolic spaces with bounded geometry and we recover their singular set as the Gromov boundary but also as the Marti…
The study shows that certain metrics on spheres prevent stable tangent cones for area-minimizing boundaries.
problem Preventing stable tangent cones for area-minimizing boundaries under specific metrics.
method Developed a perturbation theorem and used spectral theory and compactness arguments.
result A residual set of metrics on Sn+1 precludes linearly stable tangent cones for area-minimizing boundaries. This paper does not contain any new results, it is just an attempt to present, in a systematic way, one construction which establishes an interesting relationship between some ideas and notions well-known in the theory of integrable systems on Lie algebras and a rather different area of mathematics studying projectivel…
We calculate the first and the second variation formula for the sub-Riemannian area in three dimensional pseudo-hermitian manifolds. We consider general variations that can move the singular set of a C^2 surface and non-singular variation for C_H^2 surfaces. These formulas enable us to construct a stability operator fo…
Study on Dirac operator spectrum on shrinking surfaces with cusps.
problem Behavior of Dirac operator spectrum on degenerating Riemannian surfaces.
method Adapted pseudodifferential calculus, including Dirac operators and their resolvents.
result Smoothness of spectral projectors and t2logt regularity for the cusp-surgery trace. We prove lower Dirac eigenvalue bounds for closed surfaces with a spin structure whose Arf invariant equals 1. Besides the area only one geometric quantity enters in these estimates, the spin-cut-diameter which depends on the choice of spin structure. It can be expressed in terms of various distances on the surfaces or…
Generative model for Lévy area improves SDE simulation accuracy.
problem Simulating Lévy areas for high-order SDEs is challenging due to non-Gaussian nature and lack of fast sampling algorithms.
method LévyGAN, a deep-learning model with a GNN-inspired architecture, generates approximate samples of Lévy area.
result LévyGAN matches all joint and conditional odd moments exactly and achieves state-of-the-art performance in 4D Brownian motion.