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We study reflected diffusion on uniform domains where the underlying space admits a symmetric diffusion that satisfies sub-Gaussian heat kernel estimates. A celebrated theorem of Jones (Acta Math 147(1-2):71–88, 1981) states that uniform domains in Euclidean space are extension domains for Sobolev spaces. In this work, we obtain a similar extension property for metric spaces equipped with a Dirichlet form whose heat kernel satisfies a sub-Gaussian estimate. We introduce a scale-invariant version of this extension property and apply it to show that the reflected diffusion process on such a uniform domain inherits various properties from the ambient space, such as Harnack inequalities, cutoff energy inequality, and sub-Gaussian heat kernel bounds. In particular, our work extends Neumann heat kernel estimates of Gyrya and Saloff-Coste (Astérisque 336:145, 2011) beyond the Gaussian space-time scaling. Furthermore, our estimates on the extension operator imply that the energy measure of the boundary of a uniform domain is always zero. This property of the energy measure is a broad generalization of Hino's result (Probab Theory Relat Fields 156:739–793, 2013) that proves the vanishing of the energy measure on the outer square boundary of the standard Sierpiński carpet equipped with the self-similar Dirichlet form. [ABSTRACT FROM AUTHOR]

Mechanical system fault diagnosis is essential to save costs and ensure safety. Generally, rotating machinery operates in nonstationary cases, which makes fault features complex and difficult to identify. However, existing fault diagnosis methods have the following limitations. (1) Consider only time or frequency domain features fusion. (2) Extract the fusion features representation only in the Euclidean domain. Based on that, a novel method based on multisensory time-frequency features fusion and graph attention network is proposed for rotating machinery fault diagnosis under the nonstationary case. First, the multi-sensor time series are converted into multi-sensor time-frequency maps by image-to-matrix, matrix concatenation, and matrix-to-image operations. Then, simple linear iterative clustering is applied to make the superpixels in the multi-sensor time-frequency maps into nodes and form graphs based on color and texture features. Finally, the graph attention network with residual connection is applied to distinguish the rotating machinery's health status. The proposed method is verified using gearbox test rig data and bearing public data, respectively. The experimental results indicate that the proposed method can provide more reliable and accurate fault diagnosis results for rotating machinery than other methods. [ABSTRACT FROM AUTHOR]

This article, published in the Journal of Mathematical Sciences, explores the computation of the component group of an arbitrary real algebraic group. The author demonstrates that the component group is always an elementary Abelian 2-group. The computation is based on structure results on algebraic groups and Galois cohomology methods. The paper provides necessary material on algebraic groups and Galois cohomology, and includes an example of the real locus of an elliptic curve. This document discusses the computation of the group of connected components of the real locus of a connected algebraic group defined over R. It introduces the concepts of cocycles, exact sequences of groups, and Galois cohomology. The main result of the document is a formula for computing the group of connected components, which involves lattices and coroot lattices. The proof of the formula is similar to a previous proof in another paper. The given text discusses the component groups of certain mathematical structures, specifically linear algebraic groups and Abelian varieties. It presents theorems and examples related to these component groups, providing mathematical proofs and explanations. The text explores different cases and scenarios, such as when the period lattice is preserved by complex multiplication or when it is multiplied by certain values. The examples given include elliptic curves and their connected components. The text is written in a technical and mathematical language, and it may be useful for researchers studying these specific topics. [Extracted from the article]

Mappings that satisfy the inverse Poletsky inequality in a domain of the Euclidean space have been studied. Under certain conditions on the definition and mapped domains, it was found that they have a continuous extension to the boundary in terms of prime ends if the majorant involved in the Poletsky inequality is integrable over spheres. Under some additional conditions, the extension mentioned above is discrete. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Gaussian processes are popular in spatial statistics, data mining and machine learning because of their versatility in quantifying spatial variability and in propagating uncertainty. Although there has been a prolific research activity about Gaussian processes over Euclidean domains, only recently this research has extended to non-Euclidean manifolds. This paper digs into vector-valued Gaussian processes defined over the product of a hypersphere and a Euclidean space of arbitrary dimension, which are of interest in various disciplines of the natural sciences and engineering. Under mild regularity conditions, we establish a surprising one-to-one correspondence between matrix-valued kernels associated with vector Gaussian processes over the product space, and what we term partial ultraspherical and Fourier transforms that are taken over either the sphere or the Euclidean subspace. The properties of our approach are illustrated in terms of new parametric classes of matrix-valued kernels for product spaces of a hypersphere crossed with a Euclidean space. We also provide two algorithms that allow for fast simulation of approximately Gaussian (in the sense of the central limit theorem) processes in such product spaces. [ABSTRACT FROM AUTHOR]

The aim of this work is to prove two kinds of Bernstein type inequalities on certain cuspidal sets. The first one is related to the plurisubharmonic extremal function. The second type concerns star-shaped, centrally symmetric domains and uses the Euclidean distance (in directions parallel to the system axis) from the boundary. [ABSTRACT FROM AUTHOR]

Geometric properties of exceedance regions above a given quantile level provide meaningful theoretical and statistical characterizations for stochastic processes defined on Euclidean domains. Many theoretical results have been obtained for excursions of Gaussian processes and include expected values of the so-called Lipschitz–Killing curvatures (LKCs), such as the area, perimeter and Euler characteristic in two-dimensional Euclidean space. In this paper, we derive novel results for the expected LKCs of excursion sets of more general processes whose construction is based on location or scale mixtures of a Gaussian process, which means that the mean or the standard deviation, respectively, of a stationary Gaussian process is a random variable. We first present exact formulas for peaks-over-threshold-stable limit processes (so-called Pareto processes) arising from the use of Gaussian or log-Gaussian spectral functions in the spectral construction of max-stable processes. These peaks-over-threshold limits are known to arise for Gaussian location or scale mixtures if the mixing distributions satisfies certain regular-variation properties. As a second important result, we show that expected LKCs of excursion sets of such general mixture processes converge to the corresponding expressions of their Pareto process limits. We further provide exact subasymptotic formulas of expected LKCs for various specific choices of the distribution of the mixing variable. Finally, we discuss consistent empirical estimation of LKCs of exceedance regions and implement numerical experiments to illustrate the rate of convergence towards asymptotic expressions. An application to daily temperature data simulated by climate models for the period 1951–2005 over a regular pixel grid covering continental France showcases the practical utility of the new results. [ABSTRACT FROM AUTHOR]

A bosonic Laplacian is a conformally invariant second‐order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher‐order irreducible representations of the special orthogonal group. In this paper, we firstly introduce the motivation for study of the generalized Maxwell operators and bosonic Laplacians (also known as the higher spin Laplace operators). Then, with the help of connections between Rarita–Schwinger type operators and bosonic Laplacians, we solve Poisson's equation for bosonic Laplacians. A representation formula for bounded solutions to Poisson's equation in Euclidean space is also provided. In the end, we provide Green's formulas for bosonic Laplacians in scalar‐valued and Clifford‐valued cases, respectively. These formulas reveal that bosonic Laplacians are self‐adjoint with respect to a given L2 inner product on certain compact supported function spaces. [ABSTRACT FROM AUTHOR]

A triangular tile latching system consists of a set Σ of equilateral triangular tiles with at least one latchable side and an attachment rule which permits two tiles to get latched along a latchable side. In this paper we determine the language generated by a triangular tile latching system in terms of planar graphs. [ABSTRACT FROM AUTHOR]

In this paper we study an inverse boundary value problem for the biharmonic operator with first order perturbation. Our geometric setting is that of a bounded simply connected domain in the Euclidean space of dimension three or higher. Assuming that the inaccessible portion of the boundary is flat, and we have knowledge of the Dirichlet-to-Neumann map on the complement, we prove logarithmic type stability estimates for both the first and the zeroth order perturbation of the biharmonic operator. [ABSTRACT FROM AUTHOR]

This document provides a commentary on a paper titled "Marked Spatial Point Processes: Current State and Extensions to Point Processes on Linear Networks." The paper summarizes the current state of the field and proposes extensions to marked linear network point processes. It focuses on reviewing summary statistics for point processes and discusses other important topics in the field, such as marked temporal point processes and the Growth-Interaction process. The document explains different mark spaces, summary statistics, and product density/intensity functions for marked point processes. It also discusses concepts like mark filtering, marked intensity reweighted stationarity, kth-order intensity reweighted stationarity, and mark-weighted kth-order summary statistics. The text explores the use of test functions and weighted local summary statistics, as well as linear network summary statistics and marked inhomogeneous summary statistics. The authors propose proper estimators for these summary statistics and discuss their applications. [Extracted from the article]

We give an elementary approach to studying whether rings of S-integers in complex quadratic fields are Euclidean with respect to the S-norm. [ABSTRACT FROM AUTHOR]

In this paper, we find surprisingly small Oka domains in Euclidean spaces |${\mathbb {C}}^n$| of dimension |$n>1$| at the very limit of what is possible. Under a mild geometric assumption on a closed unbounded convex set |$E$| in |${\mathbb {C}}^n$|⁠ , we show that |${\mathbb {C}}^n\setminus E$| is an Oka domain. In particular, there are Oka domains only slightly bigger than a halfspace, the latter being neither Oka nor hyperbolic. This gives smooth families of real hypersurfaces |$\Sigma _t\subset {\mathbb {C}}^n$| for |$t\in {\mathbb {R}}$| dividing |${\mathbb {C}}^n$| in an unbounded hyperbolic domain and an Oka domain such that at |$t=0$|⁠ , |$\Sigma _0$| is a hyperplane and the character of the two sides gets reversed. More generally, we show that if |$E$| is a closed set in |${\mathbb {C}}^n$| for |$n>1$| whose projective closure |$\overline E\subset \mathbb {C}\mathbb {P}^n$| avoids a hyperplane |$\Lambda \subset \mathbb {C}\mathbb {P}^n$| and is polynomially convex in |$\mathbb {C}\mathbb {P}^n\setminus \Lambda \cong {\mathbb {C}}^n$|⁠ , then |${\mathbb {C}}^n\setminus E$| is an Oka domain. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We prove that all imaginary abelian sextic number fields of unit rank less than 3 and having class number 1 are Euclidean. [ABSTRACT FROM AUTHOR]

We discuss a system of third order PDEs for strictly convex smooth functions on domains of Euclidean space. We argue that it may be understood as the closure of the first order prolongation of a family of PDEs. We describe explicitly its real analytic solutions and all the solutions which satisfy a genericity condition; we also describe a family of non-generic solutions which has an application to Poisson geometry and Kähler structures on toric varieties. Our methods are geometric: we use the theory of Hessian metrics and symmetric spaces to link the analysis of the system of PDEs with properties of the manifold of matrices with orthogonal columns. [ABSTRACT FROM AUTHOR]

In this work, we introduce a family of methods for the analysis of data observed at locations scattered in three‐dimensional (3D) domains, with possibly complicated shapes. The proposed family of methods includes smoothing, regression, and functional principal component analysis for functional signals defined over (possibly nonconvex) 3D domains, appropriately complying with the nontrivial shape of the domain. This constitutes an important advance with respect to the literature, because the available methods to analyze data observed in 3D domains rely on Euclidean distances, which are inappropriate when the shape of the domain influences the phenomenon under study. The common building block of the proposed methods is a nonparametric regression model with differential regularization. We derive the asymptotic properties of the methods and show, through simulation studies, that they are superior to the available alternatives for the analysis of data in 3D domains, even when considering domains with simple shapes. We finally illustrate an application to a neurosciences study, with neuroimaging signals from functional magnetic resonance imaging, measuring neural activity in the gray matter, a nonconvex volume with a highly complicated structure. [ABSTRACT FROM AUTHOR]

Our purpose in this paper is to carry out a comprehensive investigation into the question of existence and non-existence of large solutions to semilinear elliptic equations with non-linear gradient terms in bounded domains of the Euclidean space. Based on new Keller-Osserman type integrability criteria, we provide necessary and sufficient conditions for the existence of large solutions. The novelty in our study rests on the fact that the Keller-Osserman type conditions we employ take both the growth rates of the gradient term as well as those of the first order non-linearity terms into account. When the gradient term does not appear, our criteria reduces to the well-known Keller-Osserman condition. [ABSTRACT FROM AUTHOR]

Two Kähler manifolds are called relatives if they admit a common Kähler submanifold with the same induced metrics. In this paper, we show that a Hartogs domain over an irreducible bounded symmetric domain equipped with the Bergman metric is not a relative to the complex Euclidean space. This generalizes the results in Cheng and Hao [On the non-existence of common submanifolds of Kähler manifolds and complex space forms. Ann Glob Anal Geom. 2021;60(1):167–180] and Cheng X, Niu Y. Submanifolds of Cartan–Hartogs domains and complex Euclidean spaces. J Math Anal Appl 2017;452(2):1262–1268.] and the novelty here is that the Bergman kernel of the Hartogs domain is not necessarily Nash algebraic. [ABSTRACT FROM AUTHOR]

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 | ≥ 1 2 Δ | 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. [ABSTRACT FROM AUTHOR]

In this paper, we obtain two-sided estimates for the Euclidean moment of inertia I2(G) of a convex domain G on the plane in terms of geometric characteristics of this domain similar to the Pólya–Szegő and Makai inequalities for the torsional rigidity. [ABSTRACT FROM AUTHOR]

In 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. [ABSTRACT FROM AUTHOR]

We obtain local pointwise second derivative estimates for W^{2,p}-strong solutions to a class of fully nonlinear elliptic equations on Euclidean domains, motivated by problems in conformal geometry. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider three families of groups: the Bianchi groups S L (2 , O) where O is the ring of integers of an imaginary, quadratic field; the groups S L ‡ (2 , O) where O is a ‡ -order of a definite, rational quaternion algebra with an orthogonal involution; and the groups S L (2 , O) where O is an order of a definite, rational quaternion algebra. We show that such groups are generated by elementary matrices if and only if O is semi-Euclidean (or ‡ -semi-Euclidean), which is a generalization of the usual notion of a Euclidean ring. The proofs are surprisingly simple and proceed by considering fundamental domains of Kleinian groups. [ABSTRACT FROM AUTHOR]

Random fields are widely used to characterize the inherent spatial uncertainty in engineering applications. However, for widely existing manifolds, few studies focus on the simulation of Gaussian/non‐Gaussian fields over manifolds due to the geometry complexity. Recently, we proposed a two‐stage method to simulate Gaussian fields over manifolds. In this study, we extend it to a three‐stage method for non‐Gaussian fields. In the first stage, the original manifold is dimensional reduced to a Euclidean domain by the isometric feature mapping. By this step, the geodesic distance is preserved to guarantee the correlation structure unchanged. In the second stage, conventional representations can be used to simulate the underlying Gaussian field over the Euclidean domain. In the third stage, an enhanced translation procedure is proposed to translate the underlying Gaussian field to the target non‐Gaussian field over the manifold. To explain the applicability and accuracy of the three‐stage method, two kinds of non‐Gaussian fields over three different manifolds are simulated. Then, a global error is proposed to evaluate the performance of the method, and the error analysis demonstrates the method is accurate enough. Finally, two engineering structures, that is, a compressed half‐cylinder and a Canopy shell with complex geometry, are investigated to illustrate the potential applications of the method. [ABSTRACT FROM AUTHOR]

We consider the motion of N rigid bodies – compact sets (S ε 1 , ⋯ , S ε N) ε > 0 – immersed in a viscous incompressible fluid contained in a domain in the Euclidean space R d , d = 2 , 3. We show the fluid flow is not influenced by the presence of the infinitely many bodies in the asymptotic limit ε → 0 and N = N (ε) → ∞ as soon as diam S ε i → 0 as ε → 0 , i = 1 , ⋯ , N (ε). The result depends solely on the geometry of the bodies and is independent of their mass densities. Collisions are allowed and the initial data are arbitrary with finite energy. [ABSTRACT FROM AUTHOR]

Let Ω be a bounded open domain on the Euclidean space R n and Q + be the set of all positive rational numbers. Chen and Zeng (Cal Var Part Differ Equ 56:131, 2017) investigated the eigenvalues with higher order of the fractional Laplacian (- Δ) s Ω for s > 0 and s ∈ Q + , and they obtained a universal inequality of Yang type. In the spirit of Chen and Zeng's work, we study the eigenvalues of fractional Laplacian and establish an inequality of eigenvalues with lower order under the same condition. Also, our eigenvalue inequality is universal and generalizes the eigenvalue inequality for the poly-harmonic operators given by Jost et al. (Trans Am Math Soc 363(4):1821–1854, 2011). [ABSTRACT FROM AUTHOR]

We consider the inverse problem of determining a potential in a semilinear elliptic equation from the knowledge of the Dirichlet-to-Neumann map. For bounded Euclidean domains we prove that the potential is uniquely determined by the Dirichlet-to-Neumann map measured at a single boundary point, or integrated against a fixed measure. This result is valid even when the Dirichlet data is only given on a small subset of the boundary. We also give related uniqueness results on Riemannian manifolds. [ABSTRACT FROM AUTHOR]

We investigate the existence or non-existence of spectral minimal partitions of unbounded metric graphs, where the operator applied to each of the partition elements is a Schrödinger operator of the form -Δ + V with suitable (electric) potential V, which is taken as a fixed, underlying function on the whole graph. We show that there is a strong link between spectral minimal partitions and infimal partition energies on the one hand, and the infimum Σ of the essential spectrum of the corresponding Schrödinger operator on the whole graph on the other. Namely, we show that for any k ∈ ..., the infimal energy among all admissible k-partitions is bounded from above by Σ, and if it is strictly below Σ, then a spectral minimal k-partition exists. We illustrate our results with several examples of existence and non-existence of minimal partitions of unbounded and infinite graphs, with and without potentials. The nature of the proofs, a key ingredient of which is a version of the characterization of the infimum of the essential spectrum known as Persson's theorem for quantum graphs, strongly suggests that corresponding results should hold for Schrödinger operator-based partitions of unbounded domains in Euclidean space. [ABSTRACT FROM AUTHOR]

Minkowski's First Theorem and Dirichlet's Approximation Theorem provide upper bounds on certain minima taken over lattice points contained in domains of Euclidean spaces. We study the distribution of such minima and show, under some technical conditions, that they exhibit Weibull asymptotics with respect to different natural measures on the space of unimodular lattices in \mathbb {R}^d. This follows from very general Poisson approximation results for shrinking targets which should be of independent interest. Furthermore, we show in the appendix that the logarithm laws of Kleinbock-Margulis [Invent. Math. 138 (1999), pp. 451–494], Khinchin and Gallagher [J. London Math. Soc. 37 (1962), pp. 387–390] can be deduced from our distributional results. [ABSTRACT FROM AUTHOR]

In this paper, we study the Sobolev regularity of solutions to nonlinear second order elliptic equations with super-linear first-order terms on Riemannian manifolds, complemented with Neumann boundary conditions, when the source term of the equation belongs to a Lebesgue space, under various integrability regimes. Our method is based on an integral refinement of the Bochner identity, and leads to "semilinear Calderón–Zygmund" type results. Applications to the problem of smoothness of solutions to Mean Field Games systems with Neumann boundary conditions posed on convex domains of the Euclidean space will also be discussed. [ABSTRACT FROM AUTHOR]

We study the Cauchy-Dirichlet Problem (CDP) for a nonlinear and nonlocal diffusion equation of singular type of the form $ \partial_t u = - \mathcal{L} u^m $ posed on a bounded Euclidean domain $ \Omega\subset \mathbb{R}^N $ with smooth boundary and $ N\ge 1 $. The linear diffusion operator $ \mathcal{L} $ is a sub-Markovian operator, allowed to be of nonlocal type, while the nonlinearity is of singular type, namely $ u^m = |u|^{m-1}u $ with $ 0

We study the $ L^2 $-gradient flow of functionals $ \mathscr F $ depending on the eigenvalues of Schrödinger potentials $ V $ for a wide class of differential operators associated with closed, symmetric, and coercive bilinear forms, including the case of all the Dirichlet forms (such as for second order elliptic operators in Euclidean domains or Riemannian manifolds).We suppose that $ \mathscr F $ arises as the sum of a $ (-\theta) $-convex functional $ \mathscr K $ with proper domain $ {\mathbb K}\subset L^2 $, forcing the admissible potentials to stay above a constant $ V_{\rm min} $, and a term $ {\mathscr{H}}(V) = \varphi(\lambda_1(V), \cdots, \lambda_ J(V)) $ which depends on the first $ J $ eigenvalues associated with $ V $ through a $ {\mathrm C}^1 $ function $ \varphi $.Even though $ {\mathscr{H}} $ is not a smooth perturbation of a convex functional (and it is in fact concave in simple important cases as the sum of the first $ J $ eigenvalues) and we do not assume any compactness of the sublevels of $ \mathscr K $, we prove the convergence of the Minimizing Movement method to a solution $ V\in H^1(0, T;L^2) $ of the differential inclusion $ V'(t)\in -\partial_L^- \mathscr F(V(t)) $, which under suitable compatibility conditions on $ \varphi $ can be written as$ V'(t)+\sum\limits_{i = 1}^ J\partial_i\varphi(\lambda_1(V(t)), \dots, \lambda_ J(V(t)))u_i^2(t)\in -\partial_F^- \mathscr K(V(t)) $where $ (u_1(t), \dots, u_ J(t)) $ is an orthonormal system of eigenfunctions associated with the eigenvalues $ (\lambda_1(V(t)), , \dots, \lambda_ J(V(t))) $ and $ \partial^-_L $ (resp. $ \partial^-_F $) denotes the limiting (resp. Fréchet) subdifferential. [ABSTRACT FROM AUTHOR]

Let Ω and Ω ˜ be domains in the Euclidean space ℝn. We study the boundary behavior of weak (p, q)-quasiconformal mappings φ : Ω → Ω ˜ , n – 1 < q ≤ p < n. The suggested method is based on the capacitary distortion properties of the weak (p, q)-quasiconformal mappings. [ABSTRACT FROM AUTHOR]

We establish theorems on embeddings and compact embeddings of spaces of functions of positive smoothness defined on a Hölder domain of the -dimensional Euclidean space in Lebesgue spaces. [ABSTRACT FROM AUTHOR]

We study the geometric significance of Leinster's notion of magnitude for a compact metric space. For a smooth, compact domain X in an odd-dimensional Euclidean space, we show that the asymptotic expansion of the function \mathcal {M}_X(R) = \mathrm {Mag}(R\cdot X) at R = \infty determines the Willmore energy of the boundary \partial X. This disproves the Leinster-Willerton conjecture for a compact convex body in odd dimensions. [ABSTRACT FROM AUTHOR]

Sobolev embeddings into Orlicz spaces on domains in the Euclidean space or, more generally, on Riemannian manifolds are considered. Highly irregular domains where the optimal degree of integrability of a function may be lower than the one of its gradient are focused. A necessary and sufficient condition for the validity of the relevant embeddings is established in terms of the isocapacitary function of the domain. Compact embeddings are discussed as well. Sufficient conditions involving the isoperimetric function of the domain are derived as a by-product. [ABSTRACT FROM AUTHOR]

For the spectral fractional diffusion operator of order 2s, s ∈ (0 , 1) , in bounded, curvilinear polygonal domains Ω ⊂ R 2 we prove exponential convergence of two classes of hp discretizations under the assumption of analytic data (coefficients and source terms, without any boundary compatibility), in the natural fractional Sobolev norm H s (Ω) . The first hp discretization is based on writing the solution as a co-normal derivative of a 2 + 1 -dimensional local, linear elliptic boundary value problem, to which an hp-FE discretization is applied. A diagonalization in the extended variable reduces the numerical approximation of the inverse of the spectral fractional diffusion operator to the numerical approximation of a system of local, decoupled, second order reaction-diffusion equations in Ω . Leveraging results on robust exponential convergence of hp-FEM for second order, linear reaction diffusion boundary value problems in Ω , exponential convergence rates for solutions u ∈ H s (Ω) of L s u = f follow. Key ingredient in this hp-FEM are boundary fitted meshes with geometric mesh refinement towards ∂ Ω . The second discretization is based on exponentially convergent numerical sinc quadrature approximations of the Balakrishnan integral representation of L - s combined with hp-FE discretizations of a decoupled system of local, linear, singularly perturbed reaction-diffusion equations in Ω . The present analysis for either approach extends to (polygonal subsets M ~ of) analytic, compact 2-manifolds M , parametrized by a global, analytic chart χ with polygonal Euclidean parameter domain Ω ⊂ R 2 . Numerical experiments for model problems in nonconvex polygonal domains and with incompatible data confirm the theoretical results. Exponentially small bounds on Kolmogorov n-widths of solution sets for spectral fractional diffusion in curvilinear polygons and for analytic source terms are deduced. [ABSTRACT FROM AUTHOR]

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In this paper, we present a full generalization for the power GCDQ and LCMQ matrices defined on q-ordered gcd-closed sets over Euclidean Domains. Structure theorems, determinants, reciprocals, inverses, and p-norms are also presented. In addition, Some examples are given in the Euclidean domain Z[i]. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Driven by the outstanding performance of neural networks in the structured euclidean domain, recent years have seen a surge of interest in developing neural networks for graphs and data supported on graphs. The graph is leveraged at each layer of the neural network as a parameterization to capture detail at the node level with a reduced number of parameters and computational complexity. Following this rationale, this paper puts forth a general framework that unifies state-of-the-art graph neural networks (GNNs) through the concept of EdgeNet. An EdgeNet is a GNN architecture that allows different nodes to use different parameters to weigh the information of different neighbors. By extrapolating this strategy to more iterations between neighboring nodes, the EdgeNet learns edge- and neighbor-dependent weights to capture local detail. This is a general linear and local operation that a node can perform and encompasses under one formulation all existing graph convolutional neural networks (GCNNs) as well as graph attention networks (GATs). In writing different GNN architectures with a common language, EdgeNets highlight specific architecture advantages and limitations, while providing guidelines to improve their capacity without compromising their local implementation. For instance, we show that GCNNs have a parameter sharing structure that induces permutation equivariance. This can be an advantage or a limitation, depending on the application. In cases where it is a limitation, we propose hybrid approaches and provide insights to develop several other solutions that promote parameter sharing without enforcing permutation equivariance. Another interesting conclusion is the unification of GCNNs and GATs —approaches that have been so far perceived as separate. In particular, we show that GATs are GCNNs on a graph that is learned from the features. This particularization opens the doors to develop alternative attention mechanisms for improving discriminatory power. [ABSTRACT FROM AUTHOR]

This work presents formal correctness proofs in Isabelle/HOL of algorithms to transform a matrix into Smith normal form, a canonical matrix form, in a general setting: the algorithms are written in an abstract form and parameterized by very few simple operations. We formally show their soundness provided the operations exist and satisfy some conditions, which always hold on Euclidean domains. We also provide a formal proof on some results about the generality of such algorithms as well as the uniqueness of the Smith normal form. Since Isabelle/HOL does not feature dependent types, the development is carried out by switching conveniently between two different existing libraries by means of the lifting and transfer package and the use of local type definitions, a sound extension to HOL. [ABSTRACT FROM AUTHOR]

In this paper, we study sharp two-sided heat kernel estimates for a large class of symmetric reflected diffusions with jumps on the closure of an inner uniform domain D in a length metric space. The length metric is the intrinsic metric of a strongly local Dirichlet form. When D is an inner uniform domain in the Euclidean space, a prototype for a special case of the processes under consideration is symmetric reflected diffusions with jumps on D, whose infinitesimal generators are non-local (pseudo-differential) operators \mathcal {L} on D of the form \[ \mathcal {L} u(x)\! =\!\frac 12 \!\sum _{i, j=1}^d\! \frac {\partial }{\partial x_i}\! \left (\!\!a_{ij}(x) \frac {\partial u(x)}{\partial x_j}\!\right) \!+ \lim _{\varepsilon \downarrow 0}\! \int _{\{y\in D: \, \rho _D(y, x)>\varepsilon \}}\!\! (u(y)-u(x)) J(x, y)\, dy \] satisfying "Neumann boundary condition". Here, \rho _D(x,y) is the length metric on D, A(x)=(a_{ij}(x))_{1\leq i,j\leq d} is a measurable d\times d matrix-valued function on D that is uniformly elliptic and bounded, and \[ J(x,y)≔\frac {1}{\Phi (\rho _D(x,y))} \int _{[\alpha _1, \alpha _2]} \frac {c(\alpha, x,y)} {\rho _D(x,y)^{d+\alpha }} \,\nu (d\alpha), \] where \nu is a finite measure on [\alpha _1, \alpha _2] \subset (0, 2), \Phi is an increasing function on [ 0, \infty) with c_1e^{c_2r^{\beta }} \le \Phi (r) \le c_3 e^{c_4r^{\beta }} for some \beta \in [0,\infty ], and c(\alpha, x, y) is a jointly measurable function that is bounded between two positive constants and is symmetric in (x, y). [ABSTRACT FROM AUTHOR]

Graph convolutional neural networks (GCNNs) have been successfully applied to a wide range of problems, including low-dimensional Euclidean structural domains representing images, videos, and speech and high-dimensional non-Euclidean domains, such as social networks and chemical molecular structures. However, in computer vision, the existing GCNNs are not provided with positional information to distinguish between graphs of new structures; therefore, the performance of the image classification domain represented by arbitrary graphs is significantly poor. In this work, we introduce how to initialize the positional information through a random walk algorithm and continuously learn the additional position-embedded information of various graph structures represented over the superpixel images we choose for efficiency. We call this method the graph convolutional network with learnable positional embedding applied on images (IMGCN-LPE). We apply IMGCN-LPE to three graph convolutional models (the Chebyshev graph convolutional network, graph convolutional network, and graph attention network) to validate performance on various benchmark image datasets. As a result, although not as impressive as convolutional neural networks, the proposed method outperforms various other conventional convolutional methods and demonstrates its effectiveness among the same tasks in the field of GCNNs. [ABSTRACT FROM AUTHOR]

We establish some equivalent conditions for a homeomorphism φ : D → D ′ of Euclidean domains in R n , n ≥ 2 , to induce a bounded composition operator φ ∗ : L p 1 (D ′ ; ω) ∩ Lip l (D ′) → L q 1 (D ; θ) , where 1 < q ≤ p < ∞ , by the composition rule: φ ∗ (f) = f ∘ φ . Here ω : D ′ → (0 , ∞) is an arbitrary weight function on the domain D ′ , and θ : D → (0 , ∞) is some weight function in Muckenhoupt's A q -class on the domain D. Moreover, we prove that the class of homeomorphisms under consideration is completely determined by the controlled variation of the weighted capacity of cubical condensers whose shells are concentric cubes. [ABSTRACT FROM AUTHOR]