Euclidean Domains Research Articles

Kiselman minimum principle and rooftop envelopes in complex Hessian equations

We initiate the study of m-subharmonic functions with respect to a semipositive (1, 1)-form in Euclidean domains, providing a significant element in understanding geodesics within the context of complex Hessian equations. Based on the foundational Perron envelope construction, we prove a decomposition of m-subharmonic solutions, and a general comparison principle that effectively manages singular Hessian measures. Additionally, we establish a rooftop equality and an analogue of the Kiselman minimum principle, which are crucial ingredients in establishing a criterion for geodesic connectivity among m-subharmonic functions, expressed in terms of their asymptotic envelopes.

Multilevel approximation of Gaussian random fields: Covariance compression, estimation, and spatial prediction

The distribution of centered Gaussian random fields (GRFs) indexed by compacta such as smooth, bounded Euclidean domains or smooth, compact and orientable manifolds is determined by their covariance operators. We consider centered GRFs given as variational solutions to coloring operator equations driven by spatial white noise, with an elliptic self-adjoint pseudodifferential coloring operator from the Hörmander class. This includes the Matérn class of GRFs as a special case. Using biorthogonal multiresolution analyses on the manifold, we prove that the precision and covariance operators, respectively, may be identified with bi-infinite matrices and finite sections may be diagonally preconditioned rendering the condition number independent of the dimension p of this section. We prove that a tapering strategy by thresholding applied on finite sections of the bi-infinite precision and covariance matrices results in optimally numerically sparse approximations. That is, asymptotically only linearly many nonzero matrix entries are sufficient to approximate the original section of the bi-infinite covariance or precision matrix using this tapering strategy to arbitrary precision. The locations of these nonzero matrix entries can be determined a priori. The tapered covariance or precision matrices may also be optimally diagonally preconditioned. Analysis of the relative size of the entries of the tapered covariance matrices motivates novel, multilevel Monte Carlo (MLMC) oracles for covariance estimation, in sample complexity that scales log-linearly with respect to the number p of parameters. In addition, we propose and analyze novel compressive algorithms for simulating and kriging of GRFs. The complexity (work and memory vs. accuracy) of these three algorithms scales near-optimally in terms of the number of parameters p of the sample-wise approximation of the GRF in Sobolev scales.

Polyhedral approximation and uniformization for non-length surfaces

We prove that any metric surface (that is, metric space homeomorphic to a 2 -manifold with boundary) with locally finite Hausdorff 2 -measure is the Gromov–Hausdorff limit of polyhedral surfaces with controlled geometry. We use this result, together with the classical uniformization theorem, to prove that any metric surface homeomorphic to the 2 -sphere with finite Hausdorff 2 -measure admits a weakly quasiconformal parametrization by the Riemann sphere, answering a question of Rajala–Wenger. These results have been previously established by the authors under the assumption that the metric surface carries a length metric. As a corollary, we obtain new proofs of the uniformization theorems of Bonk–Kleiner for quasispheres and of Rajala for reciprocal surfaces. Another corollary is a simplification of the definition of a reciprocal surface, which answers a question of Rajala concerning minimal hypotheses under which a metric surface is quasiconformally equivalent to a Euclidean domain.

Visibility graph-based covariance functions for scalable spatial analysis in non-convex partially Euclidean domains.

We present a new method for constructing valid covariance functions of Gaussian processes for spatial analysis in irregular, non-convex domains such as bodies of water. Standard covariance functions based on geodesic distances are not guaranteed to be positive definite on such domains, while existing non-Euclidean approaches fail to respect the partially Euclidean nature of these domains where the geodesic distance agrees with the Euclidean distances for some pairs of points. Using a visibility graph on the domain, we propose a class of covariance functions that preserve Euclidean-based covariances between points that are connected in the domain while incorporating the non-convex geometry of the domain via conditional independence relationships. We show that the proposed method preserves the partially Euclidean nature of the intrinsic geometry on the domain while maintaining validity (positive definiteness) and marginal stationarity of the covariance function over the entire parameter space, properties which are not always fulfilled by existing approaches to construct covariance functions on non-convex domains. We provide useful approximations to improve computational efficiency, resulting in a scalable algorithm. We compare the performance of our method with those of competing state-of-the-art methods using simulation studies on synthetic non-convex domains. The method is applied to data regarding acidity levels in the Chesapeake Bay, showing its potential for ecological monitoring in real-world spatial applications on irregular domains.

Formalizing Factorization on Euclidean Domains and Abstract Euclidean Algorithms

Formalizing Factorization on Euclidean Domains and Abstract Euclidean Algorithms

Supersymmetric Wormholes in String Theory.

We construct a large family of Euclidean supersymmetric wormhole solutions of type IIB supergravity which are asymptotically AdS_{5}×S^{5}. The solutions are constructed using consistent truncation to maximally gauged supergravity in five dimensions which is further truncated to a four scalar model. Within this model we perform a full analytic classification of supersymmetric domain wall solutions with flat Euclidean domain wall slices. On each side of the wormhole, the solution asymptotes to AdS_{5} dual to N=4 supersymmetric Yang-Mills deformed by a supersymmetric mass term.

Generalized finite difference method on unknown manifolds

In this paper, we extend the Generalized Finite Difference Method (GFDM) on unknown compact submanifolds of the Euclidean domain, identified by randomly sampled data that (almost surely) lie on the interior of the manifolds. Theoretically, we formalize GFDM by exploiting a representation of smooth functions on the manifolds with Taylor's expansions of polynomials defined on the tangent bundles. We illustrate the approach by approximating the Laplace-Beltrami operator, where a stable approximation is achieved by a combination of Generalized Moving Least-Squares algorithm and novel linear programming that relaxes the diagonal-dominant constraint for the estimator to allow for a feasible solution even when higher-order polynomials are employed. We establish the theoretical convergence of GFDM in solving Poisson PDEs and numerically demonstrate the accuracy on simple smooth manifolds of low and moderate high co-dimensions as well as unknown 2D surfaces. For the Dirichlet Poisson problem where no data points on the boundaries are available, we employ GFDM with the volume-constraint approach that imposes the boundary conditions on data points close to the boundary. When the location of the boundary is unknown, we introduce a novel technique to detect points close to the boundary without needing to estimate the distance of the sampled data points to the boundary. We demonstrate the effectiveness of the volume-constraint employed by imposing the boundary conditions on the data points detected by this new technique compared to imposing the boundary conditions on all points within a certain distance from the boundary, where the latter is sensitive to the choice of truncation distance and requires the knowledge of the boundary location.

IDEALS AND ALMOST PRINCIPAL IDEALS IN EUCLIDEAN Γ−SEMIRINGS

The generalization of the ring of ordinary integers and their properties into an Euclidean ring is well known. Every ideal in an Euclidean ring is a principal ideal, as is also widely known. That is, the Euclidean ring is a principal ideal ring. This paper aims to generalize the Γ−semiring of non-negative integers and their properties by defining Euclidean Γ−semiring. A Euclidean Γ−semiring is one of the many special classes of Γ−semirings. Finally, the special class of Γ−semirings discussed in this paper is the class of almost principal ideal Γ−semirings.

The Cauchy–Dirichlet problem for the fast diffusion equation on bounded domains

The Fast Diffusion Equation (FDE) ut=Δum, with m∈(0,1), is an important model for singular nonlinear (density dependent) diffusive phenomena. Here, we focus on the Cauchy–Dirichlet problem posed on smooth bounded Euclidean domains. In addition to its physical relevance, there are many aspects that make this equation particularly interesting from the pure mathematical perspective. For instance: mass is lost and solutions may extinguish in finite time, merely integrable data can produce unbounded solutions, classical forms of Harnack inequalities (and other regularity estimates) fail to be true, etc.In this paper, we first provide a survey (enriched with an extensive bibliography) focussing on the more recent results about existence, uniqueness, boundedness and positivity (i.e., Harnack inequalities, both local and global), and higher regularity estimates (also up to the boundary and possibly up to the extinction time). We then prove new global (in space and time) Harnack estimates in the subcritical regime. In the last section, we devote a special attention to the asymptotic behaviour, from the first pioneering results to the latest sharp results, and we present some new asymptotic results in the subcritical case.

Exact flow equation for the divergence functional

An exact functional renormalization group flow equation is derived for the divergence functional which is a generalization of the Kullback-Leibler divergence to quantum field theories in the Euclidean domain. It compares distributions with different sources and field expectation values. The renormalization group flow for a regularized version of this functional connects two limits: one where the functional is known in terms of the microscopic action or probability distribution, and the other where all fluctuations are taken into account. In the latter limit one can obtain full correlation functions from functional derivatives of the divergence functional. The flow equation provides a possibility to determine this functional non-perturbatively.

Euclidean Domain in the Ring Q[\(\sqrt{-43}\)

An entire ring R with unity is said to be Euclidean Domain (ED) if on R, we defined a function N: R \(\to\) \(\mathbb{Z}\)+ which admits proper generalization of the Euclidean division of integers. Every Euclidean domain (ED) is a Principal ideal domain (PID), but not all principal ideals are Euclidean. We provide detailed proof that the quadratic algebraic integer ring Q[\(\sqrt{-43}\)] is not Euclidean domain. We proved that the ring of algebraic integer in the quadratic complex field Q[\(\sqrt{-43}\)] is a principal ideal domain using the developed inequalities and field norm axioms in [1]. We proved that the ring Q[\(\sqrt{-43}\)] fails to have universal side divisors, thus, fails to be Euclidean domain (ED). This article extended the result application of [1] proving that ring R of algebraic integer in complex quadratic fields Q[\(\sqrt{-M}\)] for M = 43 is non-Euclidean PID in an understandable manner. We hope to look into the formation of these rings, thus, non-Euclidean geometries where the practical application will be more useful. E.g., Elliptic curves on finite fields.

The boundary of a graph and its isoperimetric inequality

We define, for any graph G=(V,E), a boundary ∂G⊆V. The definition coincides with what one would expected for the discretization of (sufficiently nice) Euclidean domains and contains all vertices from the Chartrand, Erwin, Johns & Zhang boundary. Moreover, it satisfies an isoperimetric principle stating that graphs with many vertices have a large boundary unless they contain long paths: we show that for graphs with maximaldegree Δ |∂G|≥12Δ|V|diam(G).For graphs discretizing Euclidean domains, one has diam(G)∼|V|1/d and recovers the scaling of the classical Euclidean isoperimetric principle.

Third homology of SL2 over number fields: The norm-Euclidean quadratic imaginary case

In the article The third homology ofSL2(Q) ([7]), Hutchinson determined the structure of H3(SL2(Q),Z[12]) by expressing it in terms of K3ind(Q)≅Z/24 and the scissor congruence group of the residue field Fp with p a prime number. In this paper, we develop further the properties of the refined scissors congruence group in order to extend this result to the case of imaginary quadratic number fields whose ring of integers is a Euclidean domain with respect to the norm.

Pólya’s conjecture for Euclidean balls

The celebrated Pólya’s conjecture (1954) in spectral geometry states that the eigenvalue counting functions of the Dirichlet and Neumann Laplacian on a bounded Euclidean domain can be estimated from above and below, respectively, by the leading term of Weyl’s asymptotics. Pólya’s conjecture is known to be true for domains which tile Euclidean space, and, in addition, for some special domains in higher dimensions. In this paper, we prove Pólya’s conjecture for the disk, making it the first non-tiling planar domain for which the conjecture is verified. We also confirm Pólya’s conjecture for arbitrary planar sectors, and, in the Dirichlet case, for balls of any dimension. Along the way, we develop the known links between the spectral problems in the disk and certain lattice counting problems. A key novel ingredient is the observation, made in recent work of the last named author, that the corresponding eigenvalue and lattice counting functions are related not only asymptotically, but in fact satisfy certain uniform bounds. Our proofs are purely analytic, except for a rigorous computer-assisted argument needed to cover the short interval of values of the spectral parameter in the case of the Neumann problem in the disk.

Variable exponent Sobolev spaces and regularity of domains-II

We provide necessary conditions on Euclidean domains for inclusions W1,p(·)(Ω)↪Lq(·)(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{1,p(\\cdot )}(\\Omega ) \\hookrightarrow L^{q(\\cdot )}(\\Omega ) $$\\end{document} of variable exponent Sobolev spaces. The conditions on the exponent p(·)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ p(\\cdot ) $$\\end{document} are log-Hölder and log-log-Hölder continuity, while those on the domain Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\Omega $$\\end{document} are the measure and the log measure density conditions. Restrictions on the exponents q(·)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ q(\\cdot ) $$\\end{document} and p(·)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ p(\\cdot )$$\\end{document} appearing in Górka et al. (J. Geom. Anal. 310: 7304-7319, 2021) are relaxed, improving the results obtained in that work.

Spectrum comparison on free boundary minimal submanifolds of Euclidean domains

AbstractIn this paper, using ideas of Simons, Ros, and Savo, we prove a comparison between the spectrums of the stability operator and the Hodge–Laplacian acting on differential 1‐forms on a compact minimal submanifold immersed into a Euclidean domain.

Computing eigenvalues of the Laplacian on rough domains

We prove a general Mosco convergence theorem for bounded Euclidean domains satisfying a set of mild geometric hypotheses. For bounded domains, this notion implies norm-resolvent convergence for the Dirichlet Laplacian which in turn ensures spectral convergence. A key element of the proof is the development of a novel, explicit Poincaré-type inequality. These results allow us to construct a universal algorithm capable of computing the eigenvalues of the Dirichlet Laplacian on a wide class of rough domains. Many domains with fractal boundaries, such as the Koch snowflake and certain filled Julia sets, are included among this class. Conversely, we construct a counterexample showing that there does not exist a universal algorithm of the same type capable of computing the eigenvalues of the Dirichlet Laplacian on an arbitrary bounded domain.

Equivalence of measures and asymptotically optimal linear prediction for Gaussian random fields with fractional-order covariance operators

We consider two Gaussian measures μ,μ˜ on a separable Hilbert space, with fractional-order covariance operators A−2β and A˜−2β˜, respectively, and derive necessary and sufficient conditions on A,A˜ and β,β˜>0 for I. equivalence of the measures μ and μ˜, and II. uniform asymptotic optimality of linear predictions for μ based on the misspecified measure μ˜. These results hold, e.g., for Gaussian processes on compact metric spaces. As an important special case, we consider the class of generalized Whittle–Matérn Gaussian random fields, where A and A˜ are elliptic second-order differential operators, formulated on a bounded Euclidean domain D⊂Rd and augmented with homogeneous Dirichlet boundary conditions. Our outcomes explain why the predictive performances of stationary and non-stationary models in spatial statistics often are comparable, and provide a crucial first step in deriving consistency results for parameter estimation of generalized Whittle–Matérn fields.

Nonlocal Perimeters and Curvature Flows on Graphs with Applications in Image Processing and High-Dimensional Data Classification

In this paper, we revisit the notion of perimeter on graphs, introduced in El Chakik, Elmoataz, and Desquesnes [Signal Process., 105 (2014), pp. 449–463], and we extend it to so-called inner and outer perimeters. We will also extend the notion of total variation on graphs. Thanks to the co-area formula, we show that discrete total variations can be expressed through these perimeters. Then, we propose a novel class of curvature operators on graphs that unifies both local and nonlocal mean curvature on an Euclidean domain. This leads us to translate and adapt the notion of the mean curvature flow on graphs as well as the level set mean curvature, which can be seen as approximate schemes. Finally, we exemplify the usefulness of these methods in image processing, 3D point cloud processing, and high dimensional data classification.

Unimodular Hartle-Hawking wave packets and their probability interpretation

We reexamine the Hartle-Hawking wave function from the point of view of a quantum theory which starts from the connection representation and allows for off-shell nonconstancy of $\mathrm{\ensuremath{\Lambda}}$ (as in unimodular theory), with a concomitant dual relational time variable. By translating its structures to the metric representation we find a nontrivial inner product rendering wave packets of Hartle-Hawking waves normalizable and the time evolution unitary; however, the implied probability measure differs significantly from the naive $|\ensuremath{\psi}{|}^{2}$. In contrast with the (monochromatic) Hartle-Hawking wave function, these packets form traveling waves with a probability peak describing de Sitter space, except near the bounce, where the incident and reflected waves interfere, transiently recreating the usual standing wave. Away from the bounce the packets get sharper both in metric and connection space, an apparent contradiction with Heisenberg's principle allowed by the fact that the metric is not Hermitian, even though its eigenvalues are real. Near the bounce, the evanescent wave not only penetrates into the classically forbidden region but also extends into the ${a}^{2}<0$ Euclidean domain. We work out the propagators for this theory and relate them to the standard ones. The $a=0$ point (aka the ``nothing'') is unremarkable, and in any case a wave function peaked therein is typically non-normalizable and/or implies a nonsensical probability for $\mathrm{\ensuremath{\Lambda}}$ (which the Universe would preserve forever). Within this theory it makes more sense to adopt a Gaussian state in an appropriate function of $\mathrm{\ensuremath{\Lambda}}$, and use the probability associated with the evanescent wave present near the time of the bounce as a measure of the likelihood of creation of a pair of time-symmetric semiclassical Universes.