Euclidean Domains Research Articles

Analytical representations of the spread harmonic measure density.

We study the spread harmonic measure that characterizes the spatial distribution of reaction events on a partially reactive surface. For Euclidean domains in which Brownian motion can be split into independent lateral and transverse displacements, we derive analytical formulas for the spread harmonic measure density and analyze its asymptotic behavior. This analysis is applicable to slab domains, general cylindrical domains, and a half-space. We investigate the spreading effect due to multiple reflections on the surface, and the underlying role of finite reactivity. We discuss further extensions and applications of analytical results to describe Laplacian transfer phenomena such as permeation through semipermeable membranes, secondary current distribution on partially blocking electrodes, and surface relaxation in nuclear magnetic resonance.

Using extended Derksen ideals in computational invariant theory

This paper contains three new algorithms for computing invariant rings. The first two apply to invariants of a finite group acting on a finitely generated algebra over a Euclidean ring. This may be viewed as a first step in “computational arithmetic invariant theory.” As a special case, the algorithms can compute multiplicative invariant rings. The third algorithm computes the invariant ring of a reductive group acting on a vector space, and often performs better than the algorithms known to date.The main tool upon which two of the algorithms are built is a generalized version of an ideal that was already used by Derksen in his algorithm for computing invariants of linearly reductive groups. As a further application, these so-called extended Derksen ideals give rise to invariantization maps, which turn an arbitrary ring element into an invariant.For the most part, the algorithms of this paper have been implemented.

Higher-order Sobolev embeddings and isoperimetric inequalities

Optimal higher-order Sobolev type embeddings are shown to follow via isoperimetric inequalities. This establishes a higher-order analogue of a well-known link between first-order Sobolev embeddings and isoperimetric inequalities. Sobolev type inequalities of any order, involving arbitrary rearrangement-invariant norms, on open sets in Rn, possibly endowed with a measure density, are reduced to much simpler one-dimensional inequalities for suitable integral operators depending on the isoperimetric function of the relevant sets. As a consequence, the optimal target space in the relevant Sobolev embeddings can be determined both in standard and in non-standard classes of function spaces and underlying measure spaces. In particular, our results are applied to any-order Sobolev embeddings in regular (John) domains of the Euclidean space, in Maz'ya classes of (possibly irregular) Euclidean domains described in terms of their isoperimetric function, and in families of product probability spaces, of which the Gauss space is a classical instance.

The broken ray transform in $n$ dimensions with flat reflecting boundary

We study the broken ray transform on $n$-dimensional Euclidean domains where the reflecting parts of the boundary are flat and establish injectivity and stability under certain conditions. Given a subset $E$ of the boundary $\partial \Omega$ such that $\partial \Omega \setminus E$ is itself flat (contained in a union of hyperplanes), we measure the attenuation of all broken rays starting and ending at $E$ with the standard optical reflection rule applied to $\partial \Omega \setminus E$. By localizing the measurement operator around broken rays which reflect off a fixed sequence of flat hyperplanes, we can apply the analytic microlocal approach of Frigyik, Stefanov, and Uhlmann for the ordinary ray transform by means of a local path unfolding. This generalizes the author's previous result for the square, although we can no longer treat reflections from corner points. Similar to the result for the two dimensional square, we show that the normal operator is a classical pseudo differential operator of order -1 plus a smoothing term with $C_{0}^{\infty}$ Schwartz kernel.

Entropic Approximation of Wasserstein Gradient Flows

This article details a novel numerical scheme to approximate gradient flows for optimal transport (i.e. Wasserstein) metrics. These flows have proved useful to tackle theoretically and numerically non-linear diffusion equations that model for instance porous media or crowd evolutions. These gradient flows define a suitable notion of weak solutions for these evolutions and they can be approximated in a stable way using discrete flows. These discrete flows are implicit Euler time stepping according to the Wasserstein metric. A bottleneck of these approaches is the high computational load induced by the resolution of each step. Indeed, this corresponds to the resolution of a convex optimization problem involving a Wasserstein distance to the previous iterate. Following several recent works on the approximation of Wasserstein distances, we consider a discrete flow induced by an entropic regularization of the transportation coupling. This entropic regularization allows one to trade the initial Wasserstein fidelity term for a Kulback-Leibler divergence, which is easier to deal with numerically. We show how KL proximal schemes, and in particular Dykstra's algorithm, can be used to compute each step of the regularized flow. The resulting algorithm is both fast, parallelizable and versatile, because it only requires multiplications by a Gibbs kernel. On Euclidean domains discretized on an uniform grid, this corresponds to a linear filtering (for instance a Gaussian filtering when $c$ is the squared Euclidean distance) which can be computed in nearly linear time. On more general domains, such as (possibly non-convex) shapes or on manifolds discretized by a triangular mesh, following a recently proposed numerical scheme for optimal transport, this Gibbs kernel multiplication is approximated by a short-time heat diffusion.

Diffusion determines the recurrent graph

We consider diffusion on discrete measure spaces as encoded by Markovian semigroups arising from weighted graphs. We study whether the graph is uniquely determined if the diffusion is given up to order isomorphism. If the graph is recurrent then the complete graph structure and the measure space are determined (up to an overall scaling). As shown by counterexamples this result is optimal. Without the recurrence assumption, the graph still turns out to be determined in the case of normalized diffusion on graphs with standard weights and in the case of arbitrary graphs over spaces in which each point has the same mass.These investigations provide discrete counterparts to studies of diffusion on Euclidean domains and manifolds initiated by Arendt and continued by Arendt/Biegert/ter Elst and Arendt/ter Elst.A crucial step in our considerations shows that order isomorphisms are actually unitary maps (up to a scaling) in our context.

Balls minimize trace constants in BV

Abstract Balls are shown to have the smallest optimal constant, among all admissible Euclidean domains, in Poincaré type boundary trace inequalities for functions of bounded variation with vanishing median or mean value.

Geometry of Prime End Boundary and the Dirichlet Problem for Bounded Domains in Metric Measure Spaces

In this note we study the Dirichlet problem associated with a version of prime end boundary of a bounded domain in a complete metric measure space equipped with a doubling measure supporting a Poincare inequality. We show the resolutivity of functions that are continuous on the prime end boundary and are Lipschitz regular when restricted to the subset of all prime ends whose impressions are singleton sets. We also consider a new notion of capacity adapted to the prime end boundary, and show that bounded perturbations of such functions on subsets of the prime end boundary with zero capacity are resolutive and that their Perron solutions coincide with the Perron solution of the original functions. We also describe some examples which demonstrate the efficacy of the prime end boundary approach in obtaining new results even for the classical Dirichlet problem for some Euclidean domains.

Functional Fluids on Surfaces

AbstractFluid simulation plays a key role in various domains of science including computer graphics. While most existing work addresses fluids on bounded Euclidean domains, we consider the problem of simulating the behavior of an incompressible fluid on a curved surface represented as an unstructured triangle mesh. Unlike the commonly used Eulerian description of the fluid using its time‐varying velocity field, we propose to model fluids using their vorticity, i.e., by a (time varying) scalar function on the surface. During each time step, we advance scalar vorticity along two consecutive, stationary velocity fields. This approach leads to a variational integrator in the space continuous setting. In addition, using this approach, the update rule amounts to manipulating functions on the surface using linear operators, which can be discretized efficiently using the recently introduced functional approach to vector fields. Combining these time and space discretizations leads to a conceptually and algorithmically simple approach, which is efficient, time‐reversible and conserves vorticity by construction. We further demonstrate that our method exhibits no numerical dissipation and is able to reproduce intricate phenomena such as vortex shedding from boundaries.

Interpolation Hilbert Spaces Between Sobolev Spaces

We explicitly describe all Hilbert function spaces that are interpolation spaces with respect to a given couple of Sobolev inner product spaces considered over $${\mathbb{R}^{n}}$$ or a half-space in $${\mathbb{R}^{n}}$$ or a bounded Euclidean domain with Lipschitz boundary. We prove that these interpolation spaces form a subclass of isotropic Hörmander spaces. They are parametrized with a radial function parameter which is OR-varying at + ∞ and satisfies some additional conditions. We give explicit examples of intermediate but not interpolation spaces.

Partial difference operators on weighted graphs for image processing on surfaces and point clouds.

Partial difference equations (PDEs) and variational methods for image processing on Euclidean domains spaces are very well established because they permit to solve a large range of real computer vision problems. With the recent advent of many 3D sensors, there is a growing interest in transposing and solving PDEs on surfaces and point clouds. In this paper, we propose a simple method to solve such PDEs using the framework of PDEs on graphs. This latter approach enables us to transcribe, for surfaces and point clouds, many models and algorithms designed for image processing. To illustrate our proposal, three problems are considered: (1) p -Laplacian restoration and inpainting; (2) PDEs mathematical morphology; and (3) active contours segmentation.

SCALE-INVARIANT BOUNDARY HARNACK PRINCIPLE IN INNER UNIFORM DOMAINS

We prove a scale-invariant boundary Harnack principle in inner uniform domains in the context of non-symmetric local, regular Dirichlet spaces. For inner uniform Euclidean domains, our results apply to divergence form operators that are not necessarily symmetric, and complement earlier results by H. Aikawa and A. Ancona.

Maximal Plurifinely Plurisubharmonic Functions

The main purpose of this paper is to introduce and study the notion of -maximal $\mathcal {F}$ -plurisubharmonic functions, which extends the notion of maximal plurisubharmonic functions on a Euclidean domain to an -domain of ℂ n in a natural way. Our main result is that a finite -plurisubharmonic function u on a plurifine domain U satisfies (d d c u) n = 0 if and only if u is -locally -maximal outside some pluripolar set. In particular, a finite -maximal plurisubharmonic function u satisfies (d d c u) n = 0.

On applications of Mathematica Package "FAPT" in QCD

We consider computational problems in the framework of nonpower Analityc Perturbation Theory and Fractional Analytic Perturbation Theory that are the generalization of the standard QCD perturbation theory. The singularity-free, finite couplings appear in these approaches as analytic images of the standard QCD coupling powers αvs(Q2) in the Euclidean and Minkowski domains, respectively. We provide a package "FAPT" based on the system Mathematica for QCD calculations of the images up to N3LO of renormalization group evolution. Application of these approaches to Bjorken sum rule analysis and Q2-evolution of higher twist μp−n4 is considered.

Cota Inferior para el Primer Valor Propio de Neumann

En este artículo se proporciona una cota inferior del primer valor propio de Neumann para un dominio euclídeo con simetría con respecto al origen

Extremal eigenvalues of the Laplacian on Euclidean domains and closed surfaces

We investigate properties of the sequences of extremal values that could be achieved by the eigenvalues of the Laplacian on Euclidean domains of unit volume, under Dirichlet and Neumann boundary conditions, respectively. In a second part, we study sequences of extremal eigenvalues of the Laplace-Beltrami operator on closed surfaces of unit area.

Sparse approximation of 3D shapes via spectral graph wavelets

This paper investigates the compressive representation of 3D meshes and articulates a novel sparse approximation method for 3D shapes based on spectral graph wavelets. The originality of this paper is centering on the first attempt of exploiting spectral graph wavelets in the sparse representation for 3D shape geometry. Conventional spectral mesh compression employs the eigenfunctions of mesh Laplacian as shape bases. The Laplacian eigenbases, generalizing the Fourier bases from Euclidean domain to manifold, exhibit global support and are neither efficient nor precise in representing local geometry. To ameliorate, we advocate an innovative approach to 3D mesh compression using spectral graph wavelets as dictionary to encode mesh geometry. In contrast to Laplacian eigenbases, the spectral graph wavelets are locally defined at individual vertices and can better capture local shape information in a more accurate way. Nonetheless, the multiscale spectral graph wavelets form a redundant dictionary as shape bases, therefore we formulate the compression of 3D shape as a sparse approximation problem that can be readily handled by powerful algorithms such as orthogonal matching pursuit. Various experiments demonstrate that our method is superior to the existing spectral mesh compression methods.

Mean curvature flow on graphs for image and manifold restoration and enhancement

In this paper, we propose an adaptation and a transcription of the mean curvature level set equation on the general discrete domain, a weighted graph. For this, we introduce perimeters on graphs using difference operators and define the curvature as the first variation of these perimeters. Then we propose a morphological scheme that unifies both local and nonlocal notions of mean curvature on Euclidean domains. Furthermore, this scheme allows to extend the mean curvature applications to process images, manifolds and data which can be represented by graphs.

Products of elementary and idempotent matrices over integral domains

A ring R such that invertible matrices over R are products of elementary matrices, is called (after Cohn) generalized Euclidean. Extending results proved by Ruitenburg for Bézout domains, characterizations of generalized Euclidean commutative domains are obtained, that relate them with the property that singular matrices are products of idempotent matrices. This latter property is investigated, focusing on 2×2 matrices, which is not restrictive in the context of Bézout domains. It is proved that domains R, that satisfy a suitable property of ideals called (princ), are necessarily Bézout domains if 2×2 singular matrices over R are products of idempotent matrices. The class of rings satisfying (princ) includes factorial and projective-free domains. The connection with the existence of a weak Euclidean algorithm, a notion introduced by O'Meara for Dedekind domains, is also investigated.

On a Note From Oliver Concerning Generalized Euclidean Group Rings

In an unpublished 1987 letter, Bob Oliver determined which elementary abelian 2-groups have generalized euclidean integral group rings. He produced a filtration of E 2(R) by normal subgroups, sandwiched between elementary and special linear relative groups, with layers that are second homology groups with mod-2 coefficients. His proof is presented here, with related consequences for some other finite groups.