It is a well-known and easily established fact that every Euclidean domain is also a principal ideal domain. However, the converse statement is not true, and this is usually shown by exhibiting as a counterexample the ring of algebraic integers in a certain, very specific quadratic field, and the proof that this works is quite unnatural and technical. In this article, we will present a family of counterexamples constructed using real closed fields.

Let R R be a unital ring satisfying the invariant basis number property, that every stably free R R -module is free, and that the complex of partial bases of every finite rank free module is Cohen–Macaulay. This class of rings includes every ring of stable rank 1 1 (e.g., any local, semi-local or Artinian ring), every Euclidean domain, and every Dedekind domain O S \mathcal {O}_S of arithmetic type where | S | > 1 |S| > 1 and S S contains at least one noncomplex place. Extending recent work of Galatius–Kupers–Randal-Williams and Kupers–Miller–Patzt, we prove that the sequence of general linear groups GL n ⁡ ( R ) \operatorname {GL}_n(R) satisfies slope- 1 1 homological stability with Z [ 1 / 2 ] \mathbb {Z}[1/2] -coefficients.

We all should learn in our abstract algebra classes that every Euclidean domain R has a minimal Euclidean function, ϕ R . Our short history starts with their introduction via Motzkin’s Lemma and moves onto Lenstra’s categorization of Euclidean functions in imaginary quadratic number fields. We examine computing minimal Euclidean functions in these fields, and apply them in short, easy proofs of standard results. Using the author’s simple formula for ϕ Z [ i ] , the only explicitly computable minimal Euclidean function the author knows for a number field other than Q , we apply the pre-images’ geometry to give a new elementary proof affirming Lenstra’s algebraic description of these sets. Figures illustrate the definitions and arguments.

In this paper, we investigate the properties of the boundedness of fractional integral operators Kα defined on general measure metric spaces. We study their action in Lebesgue spaces Lp(Y), Morrey spaces Lφp(Y), and extend our analysis to fractional Sobolev spaces Wα,p(Y). Using classical dyadic decomposition and the Hardy–Littlewood maximal operator, we establish sharp bounds for Kα in terms of kernel parameters and the geometric structure of the space. A significant contribution of this work is the proof that Kα is bounded from Wα,p(Y) to Lq(Y), where thus linking our operator-theoretic framework with the theory of nonlocal and fractional partial differential equations. These results provide valuable tools for studying regularity, a priori estimates, and solution mappings in nonlocal problems involving the fractional Laplacian and related operators on irregular or non- Euclidean domains.

Abstract In recent years, there has been growing interest in the field of functional neural networks. They have been proposed and studied with the aim of approximating continuous functionals defined on sets of functions on Euclidean domains. In this paper, we consider functionals defined on sets of functions on spheres. The approximation ability of deep ReLU neural networks is analyzed using an encoder-decoder framework on the unit sphere. An encoder is introduced first to accommodate the infinite-dimensional nature of the functional’s domain. It utilizes spherical harmonics to help us extract the latent finite-dimensional information of functions, which in turn facilitates in the next step of approximation analysis using fully connected neural networks. Moreover, real-world objects are frequently sampled discretely and are often corrupted by noise. Therefore, encoders with discrete inputs and those with discrete and random noise inputs are constructed, respectively. The approximation rates with different encoder structures are provided therein.

Motor imagery-based brain computer interface (MI-BCI) have been increasingly adopted in neurorehabilitation and related fields. The performance of MI-electroencephalogram (MI-EEG) decoding algorithms is central to the advancement of MI-BCI. However, current studies often lack rigorous investigation into the brain's complex network organization. Moreover, most existing methods do not incorporate the cross-frequency coupling (CFC) phenomena that occur during MI into their algorithmic designs, nor do they adequately account for how temporal dynamics across different MI stages influence decoding outcomes. To address these limitations, we propose the Dynamic Spectral-Spatial Interaction Convolution Neural Network (DSSICNN), a parameter-efficient MI-EEG decoding framework that jointly extracts temporal-spectral-spatial features. DSSICNN adopts a dual-branch parallel architecture to concurrently learn spatial representations in both Euclidean and non-Euclidean domains. It further integrates a CFC-inspired attention module to model cross-spectral interactions, followed by an additional attention mechanism that quantifies the contributions of distinct MI stages to decoding performance. DSSICNN achieves decoding performance on two public datasets that surpasses the current state-of-the-art (SOTA) under both session-dependent and session-independent settings. Beyond its empirical advantages, DSSICNN offers design insights for developing Graph Neural Network (GNN)-based MI-EEG decoding algorithms and provides a network neuroscience-inspired perspective for understanding the neurophysiological mechanisms underlying MI.

We positively answer the open question of Barlow and Bass about the convergence of renormalized effective resistance between opposite faces of Euclidean domains approximating a generalized Sierpiński carpet.

• Differentiable multi-fidelity inference: BMFIA enables gradient-based Bayesian inference for high-dimensional, non-differentiable multi-physics problems using low-fidelity model gradients. • Efficiency with minimal HF data: Accurate posterior estimates are achieved from only 100–300 HF–LF simulations with orders-of-magnitude speed-ups over HF-only inference. • Accuracy despite crude LF models: Robust posterior reconstruction is maintained even with weakly correlated or simplified low-fidelity models. High-dimensional Bayesian inverse analysis ( dim ≫ 100 ) is mostly unfeasible for computationally demanding, nonlinear physics-based high-fidelity (HF) models. Usually, the use of more efficient gradient-based inference schemes is impeded if the multi-physics models are provided by complex legacy codes. Adjoint-based derivatives are either exceedingly cumbersome to derive or nonexistent for practically relevant large-scale nonlinear and coupled multi-physics problems. Similarly, holistic automated differentiation w. r. t. primary variables of multi-physics codes is usually not yet an option and requires extensive code restructuring if not considered from the outset in the software design. This absence of differentiability further exacerbates the already present computational challenges. To overcome the existing limitations, we propose a novel inference approach called Bayesian multi-fidelity inverse analysis (BMFIA) , which leverages simpler and computationally cheaper lower-fidelity (LF) models that are designed to provide model derivatives. BMFIA learns a simple, probabilistic dependence of the LF and HF models, which is then employed in an altered likelihood formulation to statistically correct the inaccurate LF response. From a Bayesian viewpoint, this dependence represents a multi-fidelity (MF) conditional density (discriminative model). We demonstrate how this MF conditional density can be learned robustly in the small data regime from only a few HF and LF simulations (50 to 300), which would not be sufficient for naive surrogate approaches. The formulation is fully differentiable and allows the flexible design of a wide range of LF models. We demonstrate that BMFIA solves Bayesian inverse problems for scenarios that used to be prohibitive, such as finely-resolved and hence high-dimensional spatial reconstruction problems in two-dimensional Euclidean domains with static posteriors, given nonlinear and transient coupled poro-elastic media physics. We show that the resulting static MF posteriors are in excellent agreement with the (usually inaccessible) HF posteriors or ground-truth data and note that extending the framework to arbitrary three-dimensional domains is a natural and important direction for future work.

In this paper, we show a new explicit infinite family of principal domains that are not Euclidean domains.

We extend the study of inverse boundary value problems for quasilinear anisotropic conductivities from Euclidean domains to compact Riemannian manifolds with boundary. Given boundary voltage and current measurements, represented by the Dirichlet-to-Neumann (DN) map, we investigate whether the quasilinear anisotropic conductivity can be uniquely determined. Our main result establishes uniqueness for quasilinear anisotropic conductivities, where the conductivity tensor is given by a scalar function multiplied by a fixed Riemannian metric. Under natural geometric conditions, such as conformal flatness or boundary rigidity of the underlying manifold, we show that this scalar factor can be uniquely determined from the boundary measurements.

We study the gradient flow of the Allen–Cahn equation with fixed boundary contact angle in Euclidean domains for initial data with bounded energy. Under general assumptions, we establish both interior and boundary convergence properties for the solutions and associated energy measures. Under various boundary nonconcentration assumptions, we show that, for almost every time, the associated limiting varifolds satisfy generalised contact angle conditions and have bounded first variation, as well as deducing that the trace of the limit of the solutions coincides with the limit of their traces. Moreover, we derive an Ilmanen-type monotonicity formula, for initial data with bounded energy, valid for the associated energy measures up to the boundary.

Abstract The usual division algorithms on ${\mathbb {Z}}$ and ${\mathbb {Z}}[i]$ measure the size of remainders using the algebraic norm. These rings are Euclidean with respect to several functions. The pointwise minimum of all Euclidean functions $f: R \setminus \{0\} \rightarrow {\mathbb {N}}$ on a Euclidean domain R is itself a Euclidean function, called the minimal Euclidean function and denoted by $\phi _R$ . To the author’s knowledge, the integers, ${\mathbb {Z}}$ and the Gaussians, ${\mathbb {Z}}[i]$ are the only rings of integers of number fields for which we have a formula to compute their minimal Euclidean functions, $\phi _{{\mathbb {Z}}}$ and $\phi _{{\mathbb {Z}}[i]}$ . This article presents the first division algorithm (that the author knows of) for ${\mathbb {Z}}[i]$ relative to $\phi _{{\mathbb {Z}}[i]}$ , empowering readers to perform the Euclidean algorithm on ${\mathbb {Z}}[i]$ using its minimal Euclidean function.

Abstract This paper introduces the sigma flow model for the prediction of structured labelings of data observed on Riemannian manifolds, including Euclidean image domains as special case. The approach combines the Laplace–Beltrami framework for image denoising and enhancement, introduced by Sochen, Kimmel and Malladi about 25 years ago, and the assignment flow approach introduced and studied by the authors. The sigma flow arises as the Riemannian gradient flow of generalized harmonic energies and is thus governed by a nonlinear geometric PDE which determines a harmonic map from a closed Riemannian domain manifold to a statistical manifold, equipped with the Fisher–Rao metric from information geometry. A specific ingredient of the sigma flow is the mutual dependency of the Riemannian metric of the domain manifold on the evolving state. This makes the approach amenable to machine learning in a specific way, by realizing this dependency through a mapping with compact time-variant parametrization that can be learned from data. Proof-of-concept experiments demonstrate the expressivity of the sigma flow model and prediction performance. Structural similarities to transformer network architectures and networks generated by the geometric integration of sigma flows are pointed out, which highlights the connection to deep learning and, conversely, may stimulate the use of geometric design principles for structured prediction in other areas of scientific machine learning.

Almost all problems in applied mathematics, including the analysis of dynamical systems, deal with spaces of real-valued functions on Euclidean domains in their formulation and solution. In this paper, we describe the the tool Ariadne, which provides a rigorous calculus for working with Euclidean functions. We first introduce the Ariadne framework, which is based on a clean separation of objects as providing exact, effective, validated and approximate information. We then discuss the function calculus as implemented in Ariadne, including polynomial function models which are the fundamental class for concrete computations. We then consider solution of some core problems of functional analysis, namely solution of algebraic equations and differential equations, and briefly discuss their use for the analysis of hybrid systems. We will give examples of C++ and Python code for performing the various calculations. Finally, we will discuss progress on extensions, including improvements to the function calculus and extensions to more complicated classes of system.

Let (M n , g 0 ) be a smooth compact Riemannian manifold of dimension n 3 with nonempty boundary M.Let n be a symmetric convex cone and f a symmetric defining function for satisfying standard assumptions.Under an algebraic condition on , which is satisfied for example by the Grding cones + k when k < 1 2 n, we prove the existence of a locally Lipschitz viscosity solution g u = e 2u g 0 to the fully nonlinear Loewner-Nirenberg problem associated to ( f, ),where A g u is the Schouten tensor of g u .Previous results on Euclidean domains show that, in general, u is not differentiable.The solution u is obtained as the limit of smooth solutions to a sequence of fully nonlinear Loewner-Nirenberg problems on approximating cones containing (1, 0, . . ., 0), for which we also have uniqueness.In the process, we obtain an existence and uniqueness result for the corresponding Dirichlet boundary value problem with finite boundary data, which is also of independent interest.An important feature of our paper is that the existence of a conformal metric g satisfying (-g -1 A g ) on M is a consequence of our results, rather than an assumption.1. Introduction 2203 2. Proof of Theorem 1.8: the local interior gradient estimate 2210 3. Proof of Theorem 1.6: the Dirichlet boundary value problem 2217 4. Proof of Theorem 1.1 : the fully nonlinear Loewner-Nirenberg problem 2229 Appendix A. Proof of Proposition 2.4: a cone property 2237 Appendix B. The Schouten tensor for a radial conformal factor 2237

Escape-from-a-layer approach for simulating the boundary local time in Euclidean domains

A rooted network consists of a connected, locally finite graph G, equipped with edge conductances and a distinguished vertex o. A nonnegative function on the vertices of G which vanishes at o, has Laplacian 1 at o, and is harmonic at all other vertices is called a potential. We prove that every infinite recurrent rooted network admits a potential tending to infinity. This is an analogue of classical theorems due to Evans and Nakai in the settings of Euclidean domains and Riemannian surfaces.

STF: A spherical transformer for versatile cortical surfaces applications.

We prove that the reconstruction of a certain type of length spaces from their travel time data on a closed subset is Lipschitz stable. The travel time data is the set of distance functions from the entire space, measured on the chosen closed subset. The case of a Riemannian manifold with boundary with the boundary as the measurement set appears is a classical geometric inverse problem arising from Gel’fand’s inverse boundary spectral problem. Examples of spaces satisfying our assumptions include some non-simple Riemannian manifolds, Euclidean domains with non-trivial topology, and metric trees.

Abstract We provide an upper estimate for the eigenvalues of the curl curl operator on a bounded, three‐dimensional Euclidean domain in terms of eigenvalues of the Dirichlet Laplacian. The result complements recent inequalities between curl curl and Neumann Laplacian eigenvalues. The curl curl eigenvalues considered here correspond to the Maxwell eigenvalue problem with constant material parameters.