Poincare
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Related Articles from SNS
On Effective Banach-Mazur Games and an application to the Poincar\'e Recurrence Theorem for Category
arXiv:2506.11118v2 Announce Type: replace-cross Abstract: The classical Banach-Mazur game characterizes sets of first category in a topological space. In this work, we show that an effectivized version of the game yields a characterization of sets of effective first category. Using this, we give a proof for the effective Banach Category Theorem.
PJ-RoPE: A Fourier-Jet-Affine Position Space for Relative Attention
Announce Type: new Abstract: We unify RoPE's Fourier phase, Jordan-RoPE's finite jets, and ALiBi's affine recency into a single learnable relative-position space, and study which regions of this space are selected by different tasks. PJ-RoPE is a Fourier-Jet-Affine formulation for relative attention, with an optional Poincare-type reading as the affine completion of a homogeneous Fourier-jet positional representation. Algebraically, the same primitives form a finite constant-coefficient...
Well-posedness and finite element approximation of the electrostatic shear Alfv\'en wave equations
new Abstract: The aim of this paper is to study the well-posedness and finite element approximation of the electrostatic shear Alfv\'en wave equations, a coupled system of two partial differential equations arising in plasma physics as a simplified sub-model of the drift-reduced Braginskii equations. To this end, anisotropic Sobolev spaces depending on the normalized magnetic field $\b$ are introduced, together with a Poincar\'e-type inequality along the integral curves of $\b$, which holds...
Controlled generation of ultrafast vector vortex beams from a mode-locked fiber laser
Announce Type: new Abstract: We report on a new class of mode-locked fiber laser that allows direct creation of ultrafast vector vortex beams at arbitrary positions on the higher-order Poincar\'{e} sphere. The on-demand generation of space-variant polarization patterns was realized by controlling geometric phases inside the laser resonator to map polarization to orbital angular momentum. Thanks to the ingenious cavity design, the required intracavity manipulation of the geometric phase...
Unified Geometry-Guided ML-FTLE for Tracking Transient Chaos from Scalar Time Series
arXiv:2606.07385v1 Announce Type: cross Abstract: Detecting transient chaos from scalar observations without governing equations represents a fundamental challenge in nonlinear dynamics. We propose a geometry-guided machine learning framework that unifies predictive trajectory divergence with macroscopic attractor morphology to track abrupt regime shifts. The methodology extracts a local instability scale via out-of-sample k-nearest neighbor forecast errors to establish the ML-FTLE...
Unified Geometry-Guided ML-FTLE for Tracking Transient Chaos from Scalar Time Series
arXiv:2606.07385v1 Announce Type: cross Abstract: Detecting transient chaos from scalar observations without governing equations represents a fundamental challenge in nonlinear dynamics. We propose a geometry-guided machine learning framework that unifies predictive trajectory divergence with macroscopic attractor morphology to track abrupt regime shifts. The methodology extracts a local instability scale via out-of-sample k-nearest neighbor forecast errors to establish the ML-FTLE...
Expressivity of congruence-based architectures for DNNs on positive-definite matrices
new Abstract: This work studies neural architectures for classifying symmetric positive-definite matrices, focusing on congruence-like layers, in which the input matrix is multiplied on the left and right by a (possibly rectangular) weight matrix $W$ and its transpose. Such layers lie at the core of the celebrated SPDNet and have also been employed independently for dimensionality reduction on positive-definite data. We show that the (semi)-orthogonality constraint commonly imposed on $W$...
Error estimates for full discretization by an almost mass conservation technique for Cahn--Hilliard systems with dynamic boundary conditions
arXiv:2502.03847v2 Announce Type: replace Abstract: A proof of optimal-order error estimates is given for the full discretization of the bulk--surface Cahn--Hilliard system with dynamic boundary conditions in a smooth domain. The numerical method combines a linear bulk--surface finite element discretization in space and linearly implicit backward difference formulae of order one to five in time. The error estimates are obtained by a consistency and stability analysis, based on an energy...
Zeroth-Order Non-Log-Concave Sampling with Variance Reduction and Applications to Inverse Problems
arXiv:2605.30573v1 Announce Type: new Abstract: Sampling from high-dimensional, non-log-concave distributions with unnormalized densities remains a fundamental challenge in machine learning, particularly in black-box settings where gradient information is inaccessible or computationally prohibitive. While Langevin dynamics provides a principled framework for sampling when gradients are accessible, its extension to the black-box settings suffers from high variance and lacks non-asymptotic...
Quantum Cut Sparsifiers
arXiv:2606.09728v1 Announce Type: cross Abstract: In this paper, we continue a line of research initiated by Basu, Brakensiek, and Putterman [2026] studying the sparsifiability of Hamiltonians. We focus particularly on the sparsifiability of the widely-studied Quantum Cut (QC) Hamiltonians. Our main result is that in an $n$-qubit system, any $n$-qubit QC Hamiltonian can be sparsified to $\widetilde{O}(n /\varepsilon^2)$