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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]

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]

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]

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]

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]

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 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]

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]

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]

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]

In the traditional approach to differentiability, found in almost all university textbooks, this notion is considered only for interior points of the domain of function or for functions with an open domain. This approach leads to the fact that differentiability has usually been considered only for functions with an open domain in R n , which severely limits the possibility of applying the potential techniques and tools of differential calculus to a broader class of functions. Although there is a great need for generalization of the notion of differentiability of a function in various problems of mathematical analysis and other mathematical branches, the notion of differentiability of a function at the non-interior points of its domain has almost not been considered or successfully defined. In this paper, we have generalized the differentiability of scalar and vector functions of several variables by defining it at non-interior points of the domain of the function, which include not only boundary points but also all points at which the notion of linearization is meaningful (points admitting nbd rays). This generalization allows applications in all areas where standard differentiability can be applied. With this generalized approach to differentiability, some unexpected phenomena may occur, such as a function discontinuity at a point where a function is differentiable, the non-uniqueness of differentials... However, if one reduces this theory only to points with some special properties (points admitting a linearization space with dimension equal to the dimension of the ambient Euclidean space of the domain and admitting a raylike neighborhood, which includes the interior points of a domain), then all properties and theorems belonging to the known theory of differentiability remain valid in this extended theory. For generalized differentiability, the corresponding calculus (differentiation techniques) is also provided by matrices—representatives of differentials at points. In this calculus the role of partial derivatives (which in general cannot exist for differentiable functions at some points) is taken by directional derivatives. [ABSTRACT FROM AUTHOR]

Few-shot human-object interaction (FS-HOI) recognition aims at inferring new interactions between human actions and surrounding objects merely with a few available instances. It is beneficial to alleviate the long-tail and combinatorial explosion problems in human-object interaction (HOI). Nevertheless, the existing FS-HOI methods only focus on modeling the relationships between labeled samples and unlabeled samples in the Euclidean domain, which neglects the rich relational structures of the visual information among labeled samples and between human actions and objects. Accordingly, we tackle the few-shot HOI task in the non-Euclidean domain and present a graph-based model, namely, task-oriented high-order context graph network (THCG-Net). It contains a task attention module (TA-Module) and a high-order context graph module (HG-Module). In TA-Module, an attention mechanism is designed by utilizing task information to build a task-oriented space, in which the discriminative information for the current task (episode) is captured by embedding the visual features into the task-oriented space. The HG-Module is proposed to construct a task-level graph and takes the context information as high-order knowledge, which provides discriminative guidance for propagating visual information. It captures the discriminability among different categories while highlights the commonality of related categories adaptively, which effectively transfers knowledge to related categories. Extensive experimental results on two benchmark datasets, HICO-FS and TUHOI-FS, are provided. It demonstrates that our THCG-Net significantly outperforms the state-of-the-art approaches, which proves its impressive effectiveness in recognizing various human actions and surrounding objects in few-shot scenarios. [ABSTRACT FROM AUTHOR]

Let A be an n-by-n matrix with index α > 0 and b ∈ Cn. In this paper, the problem of stagnation of the DGMRES method for the singular linear system Ax = b is considered. We show that DGMRES(A, b, α) has partial stagnation of order at least k if and only if (0, . . ., 0) belongs to the the joint numerical range of matrices {Bα+1, . . ., Bα+k}, where B is a compression of A to the range of Aα. Also, we characterize the nonsingular part of a matrices A such that DGMRES(A, b, α) does not stagnate for all b ∈ Cn. Moreover, a sufficient condition for non-existence of real stagnation vectors b ∈ R(Aα) for the DGMRES method is presented, and the DGMRES stagnation of special matrices are studied. [ABSTRACT FROM AUTHOR]

Weinberger [16] in 1972, proved that the ring of integers of a number field with unit rank at least 1 is a principal ideal domain if and only if it is a Euclidean domain, provided the generalised Riemann hypothesis holds. Lenstra [13], extended the notion of Euclidean domains in order to capture Dedekind domains with finite cyclic class group and proved an analogous theorem in this setup. More precisely, he showed that the class group of the ring of integers of a number field with unit rank at least 1 is cyclic if and only if it has a Euclidean ideal class, provided the generalised Riemann hypothesis holds. The aim of this paper is to show the following. Suppose that K 1 and K 2 are two Galois number fields of odd prime degree with cyclic class groups and Hilbert class fields that are abelian over Q . If K 1 K 2 is ramified over K i , then at least one K i ( i ∈ { 1 , 2 } ) must have a Euclidean ideal class. [ABSTRACT FROM AUTHOR]

The study of certain differential operators between Sobolev spaces of sections of vector bundles on compact manifolds equipped with rough metric is closely related to the study of locally Sobolev functions on domains in the Euclidean space. In this paper, we present a coherent rigorous study of some of the properties of locally Sobolev-Slobodeckij functions that are especially useful in the study of differential operators between sections of vector bundles on compact manifolds with rough metric. The results of this type in published literature generally can be found only for integer order Sobolev spaces W m , p or Bessel potential spaces H s . Here, we have presented the relevant results and their detailed proofs for Sobolev-Slobodeckij spaces W s , p where s does not need to be an integer. We also develop a number of results needed in the study of differential operators on manifolds that do not appear to be in the literature. [ABSTRACT FROM AUTHOR]

In recent years, we have witnessed the developments that deep learning has brought to machine learning. It has solved many problems in the areas of computer vision, speech recognition, natural language processing, and various other tasks with stateof- the-art performance. However, the data in these tasks is typically represented in Euclidean space. As technology develops, more and more applications are generating data from non-Euclidean domains and representing them as graphs with complex relationships and interdependencies between objects. This poses a significant challenge to deep learning algorithms. This is because, due to the uniqueness of graphs, applying deep learning to the ubiquitous graph data is not an easy task. To solve the problem in non-Euclidean domains, Graph Neural Networks (GNNs) have emerged. A Graph Neural Network (GNN) is a neural model that captures dependencies between graphs by passing messages between graph nodes. This paper introduces commonly used graph neural networks, their learning methods, and common datasets for graph neural networks. It also provides an outlook on the future of Graph Neural Networks. [ABSTRACT FROM AUTHOR]

This article presents a graph convolutional network model (GCNM) for gait phase classification to control a lower limb exoskeleton. The GCNM can recognize four leg phases between the foot and ground, including heel strike, foot flat, heel off, and swing. The proposed model can solve the gait phase classification problem from non-Euclidean domain based on a graph mechanism for exoskeletons. Real-time performance of the gait data acquisition system was then verified on a lower limb exoskeleton. The system quantifies the time delay with an optoelectronic system, VICON BONITA 10. Comparison works with some existing state-of-the-art methods from the Euclidean domain, i.e., long short-term memory (LSTM) and deep convolutional neural network (DCNN), were also provided. Experimental works show that the proposed model has a significantly higher prediction accuracy and better robustness in gait phase classification for different people in level, uphill, and downhill environments. Moreover, the label rate is under ${\rm {10}}{\rm {\% }}$ , compared to LSTM and DCNN, which are usually at ${\rm {70}}$ – ${\rm {80}}{\rm {\% }}$. Finally, the GCNM's maximum accuracy of gait phase classification is ${\rm {97}}{\rm {.43\% }}$ ; thus, the effectiveness of the proposed model is verified. [ABSTRACT FROM AUTHOR]

The isoperimetric problem of maximizing all eigenvalues of the Laplacian on a metric tree graph within the class of trees of a given average edge length is studied. It turns out that, up to rescaling, the unique maximizer of the k-th positive eigenvalue is the star graph with three edges of lengths 2 k - 1, 1 and 1. This complements the previously known result that the first nonzero eigenvalue is maximized by all equilateral star graphs and indicates that optimizers of isoperimetric problems for higher eigenvalues may be less balanced in their shape—an observation which is known from numerical results on the optimization of higher eigenvalues of Laplacians on Euclidean domains. [ABSTRACT FROM AUTHOR]

Graph is a natural data structure for describing complex systems, which contains a set of objects and relationships. Ubiquitous real-life biomedical problems can be modeled as graph analytics tasks. Machine learning, especially deep learning, succeeds in vast bioinformatics scenarios with data represented in Euclidean domain. However, rich relational information between biological elements is retained in the non-Euclidean biomedical graphs, which is not learning friendly to classic machine learning methods. Graph representation learning aims to embed graph into a low-dimensional space while preserving graph topology and node properties. It bridges biomedical graphs and modern machine learning methods and has recently raised widespread interest in both machine learning and bioinformatics communities. In this work, we summarize the advances of graph representation learning and its representative applications in bioinformatics. To provide a comprehensive and structured analysis and perspective, we first categorize and analyze both graph embedding methods (homogeneous graph embedding, heterogeneous graph embedding, attribute graph embedding) and graph neural networks. Furthermore, we summarize their representative applications from molecular level to genomics, pharmaceutical and healthcare systems level. Moreover, we provide open resource platforms and libraries for implementing these graph representation learning methods and discuss the challenges and opportunities of graph representation learning in bioinformatics. This work provides a comprehensive survey of emerging graph representation learning algorithms and their applications in bioinformatics. It is anticipated that it could bring valuable insights for researchers to contribute their knowledge to graph representation learning and future-oriented bioinformatics studies. [ABSTRACT FROM AUTHOR]

In the stochastic finite element analysis of irregular wall structures considering material uncertainties, the random fields simulation and deterministic finite element analysis (FEA) are the two focuses. In this paper, we present an efficient stochastic finite element analysis procedure for irregular wall structures with inelastic random field properties. In the procedure, the geometry domains of irregular wall structures are deemed as two-dimensional (2D) manifolds. Then Isometric feature mapping (Isomap) is used to map the manifold to a 2D Euclidean domain, over which the well-developed stochastic harmonic function representation is applied to simulate the random field. Meanwhile, to accurately reflect the nonlinear behaviors of the irregular wall structures, we adopt an enhanced deterministic FEA method, which combines the multi-layered shell element, the softened damage-plasticity concrete model and the quasi-Newton solution with a two-level secant stiffness in a framework. Finally, the proposed approach is applied to the stochastic analysis of a U-shaped reinforced concrete shear wall to illustrate its feasibility and applicability. The results demonstrate that the proposed approach can effectively simulate the random fields over irregular geometry domains and can reproduce the representative stochastic inelastic behaviors of irregular wall structures. The approach can be further used in reliability or safety evaluation of structures. [ABSTRACT FROM AUTHOR]

This note presents some examples of programming exercises usable in a first course on abstract algebra. The topics include detecting the properties of an algebraic operation on a finite set, implementing the Euclidean algorithm in different Euclidean rings, long division of polynomials over a finite field, finding all low-degree irreducible polynomials over a finite field. [ABSTRACT FROM AUTHOR]

We show that a K-quasiregular ω -curve from a Euclidean domain to a Euclidean space with respect to a covector ω is locally (1 / K) (‖ ω ‖ / | ω | ℓ 1) -Hölder continuous. We also show that quasiregular curves enjoy higher integrability. [ABSTRACT FROM AUTHOR]

We examine Euclidean plane domains with their hyperbolic or quasihyperbolic distance. We prove that the associated metric spaces are quasisymmetrically equivalent if and only if they are bi-Lipschitz equivalent. On the other hand, for Gromov hyperbolic domains, the two corresponding Gromov boundaries are always quasisymmetrically equivalent. Surprisingly, for any finitely connected hyperbolic domain, these two metric spaces are always quasiisometrically equivalent. We construct examples where the spaces are not quasiisometrically equivalent. [ABSTRACT FROM AUTHOR]

We study parabolic equations in variable Hölder spaces on domains of Euclidean spaces. The existence and uniqueness of solutions is proved. [ABSTRACT FROM AUTHOR]

We study Leinster's notion of magnitude for a compact metric space. For a smooth, compact domain X ⊂ R²m-1, we find geometric significance in the function MX(R) = mag(R·X). The function MX extends from the positive half-line to a meromorphic function in the complex plane. Its poles are generalized scattering resonances. In the semiclassical limit R → ∞, MX admits an asymptotic expansion. The three leading terms of MX at R =+∞ are proportional to the volume, surface area and integral of the mean curvature. In particular, for convex X the leading terms are proportional to the intrinsic volumes, and we obtain an asymptotic variant of the convex magnitude conjecture by Leinster and Willerton, with corrected coefficients. [ABSTRACT FROM AUTHOR]

Extreme Learning Machine (ELM) is characterized by simplicity, generalization ability, and computational efficiency. However, previous ELMs fail to consider the inherent high-order relationship among data points, resulting in being powerless on structured data and poor robustness on noise data. This paper presents a novel semi-supervised ELM, termed Hypergraph Convolutional ELM (HGCELM), based on using hypergraph convolution to extend ELM into the non-Euclidean domain. The method inherits all the advantages from ELM, and consists of a random hypergraph convolutional layer followed by a hypergraph convolutional regression layer, enabling it to model complex intraclass variations. We show that the traditional ELM is a special case of the HGCELM model in the regular Euclidean domain. Extensive experimental results show that HGCELM remarkably outperforms eight competitive methods on 26 classification benchmarks. [ABSTRACT FROM AUTHOR]

In this paper, we study a doubly nonlinear parabolic equation arising from the gradient flow for p -Sobolev type inequality, referred as p -Sobolev flow. In the special case p = 2 our theory includes the classical Yamabe flow on a bounded domain in Euclidean space. Our main aim is to prove the global existence of the p -Sobolev flow together with its qualitative properties. [ABSTRACT FROM AUTHOR]

We study the behavior of a certain class of mappings of a domain in Euclidean space. We prove that this class is equicontinuous both at the interior and boundary points, of the domain provided that it consists of mappings that satisfy a common normalization condition and whose quasiconformality characteristic has only tempered growth in a neighborhood of each point in the closure of the domain. [ABSTRACT FROM AUTHOR]

In this paper we state and explain techniques useful for the computation of strong Gröbner and standard bases over Euclidean domains: First we investigate several strategies for creating the pair set using an idea by Lichtblau. Then we explain methods for avoiding coefficient growth using syzygies. We give an in-depth discussion on normal form computation resp. a generalized reduction process with many optimizations to further avoid large coefficients. These are combined with methods to reach GCD-polynomials at an earlier stage of the computation. Based on various examples we show that our new implementation in the computer algebra system Singular is, in general, more efficient than other known implementations. [ABSTRACT FROM AUTHOR]

We show that the only metric measure space with the structure of an N-cone and with two-sided synthetic Ricci bounds is the Euclidean space RNC1 for N integer. This is based on a novel notion of Ricci curvature upper bounds for metric measure spaces given in terms of the short time asymptotic of the heat kernel in the L2-transport distance. Moreover, we establish rigidity results of independent interest which characterize the N-dimensional standard sphere SN as the unique minimizer e.g. of among all metric measure spaces with dimension bounded above by N and Ricci curvature bounded below by N - 1. [ABSTRACT FROM AUTHOR]

Distance covariance is a measure of dependence between two random variables that take values in two, in general different, metric spaces, see Székely et al. (2007) and Lyons (2013). It is known that the distance covariance, and its generalization α-distance covariance, can be defined in several different ways that are equivalent under some moment conditions. The present paper considers four such definitions and find minimal moment conditions for each of them, together with some partial results when these conditions are not satisfied. Another purpose of the present paper is to improve existing results on consistency of distance covariance, estimated using the empirical distribution of a sample. The paper also studies the special case when the variables are Hilbert space valued, and shows under weak moment conditions that two such variables are independent if and only if their (α-) distance covariance is 0; this extends results by Lyons (2013) and Dehling et al. (2020). The proof uses a new definition of distance covariance in the Hilbert space case, generalizing the definition for Euclidean spaces using characteristic functions by Székely et al. (2007). [ABSTRACT FROM AUTHOR]

We give a short account of corollaries of the Axiom of Choice, which show the existence of curious configurations or colorings of the Euclidean spaces. The paper is readable even with a minimal knowledge of set theory. A possible new result is also proved: pairwise disjoint flags can be elected on each point of the plane. [ABSTRACT FROM AUTHOR]

Convolutional neural networks (CNNs) provide a dramatically powerful class of models, but are subject to traditional convolution that can merely aggregate permutation-ordered and dimension-equal local inputs. It causes that CNNs are allowed to only manage signals on Euclidean or grid-like domains (e.g., images), not ones on non-Euclidean or graph domains (e.g., traffic networks). To eliminate this limitation, we develop a local-aggregation function, a sharable nonlinear operation, to aggregate permutation-unordered and dimension-unequal local inputs on non-Euclidean domains. In the context of the function approximation theory, the local-aggregation function is parameterized with a group of orthonormal polynomials in an effective and efficient manner. By replacing the traditional convolution in CNNs with the parameterized local-aggregation function, Local-Aggregation Graph Networks (LAGNs) are readily established, which enable to fit nonlinear functions without activation functions and can be expediently trained with the standard back-propagation. Extensive experiments on various datasets strongly demonstrate the effectiveness and efficiency of LAGNs, leading to superior performance on numerous pattern recognition and machine learning tasks, including text categorization, molecular activity detection, taxi flow prediction, and image classification. [ABSTRACT FROM AUTHOR]

Recently it has been proved that if σ ∈ GL n (R) where R is a commutative ring and n ≥ 3 , then each of the matrices t k l (σ i j) (i ≠ j , k ≠ l) is a product of eight E n (R) -conjugates of σ and σ − 1. In this article we show that similar results hold true if R is a (noncommutative) von Neumann regular ring, or a Banach algebra, or a ring satisfying a stable range condition, or a ring with Euclidean algorithm, or an almost commutative ring. [ABSTRACT FROM AUTHOR]

Image denoising technologies in a Euclidean domain have achieved good results and are becoming mature. However, in recent years, many real-world applications encountered in computer vision and geometric modeling involve image data defined in irregular domains modeled by huge graphs, which results in the problem on how to solve image denoising problems defined on graphs. In this paper, we propose a novel model for removing mixed or unknown noise in images on graphs. The objective is to minimize the sum of a weighted fidelity term and a sparse regularization term that additionally utilizes wavelet frame transform on graphs to retain feature details of images defined on graphs. Specifically, the weighted fidelity term with $\ell _{1}$ -norm and $\ell _{2}$ -norm is designed based on a analysis of the distribution of mixed noise. The augmented Lagrangian and accelerated proximal gradient methods are employed to achieve the optimal solution to the problem. Finally, some supporting numerical results and comparative analyses with other denoising algorithms are provided. It is noted that we investigate image denoising with unknown noise or a wide range of mixed noise, especially the mixture of Poisson, Gaussian, and impulse noise. Experimental results reported for synthetic and real images on graphs demonstrate that the proposed method is effective and efficient, and exhibits better performance for the removal of mixed or unknown noise in images on graphs than other denoising algorithms in the literature. The method can effectively remove mixed or unknown noise and retain feature details of images on graphs. It delivers a new avenue for denoising images in irregular domains. [ABSTRACT FROM AUTHOR]

We consider the first eigenvalue λ 1 (Ω , σ) of the Laplacian with Robin boundary conditions on a compact Riemannian manifold Ω with smooth boundary, σ ∈ R being the Robin boundary parameter. When σ > 0 we give a positive, sharp lower bound of λ 1 (Ω , σ) in terms of an associated one-dimensional problem depending on the geometry through a lower bound of the Ricci curvature of Ω, a lower bound of the mean curvature of ∂Ω and the inradius. When the boundary parameter is negative, the lower bound becomes an upper bound. In particular, explicit bounds for mean-convex Euclidean domains are obtained, which improve known estimates. Then, we extend a monotonicity result for λ 1 (Ω , σ) obtained in Euclidean space by Giorgi and Smits [10] , to a class of manifolds of revolution which include all space forms of constant sectional curvature. As an application, we prove that λ 1 (Ω , σ) is uniformly bounded below by (n − 1) 2 4 for all bounded domains in the hyperbolic space of dimension n , provided that the boundary parameter σ ≥ n − 1 2 (McKean-type inequality). Asymptotics for large hyperbolic balls are also discussed. [ABSTRACT FROM AUTHOR]

A new cognitive approach is proposed for understanding the texts of planimetric tasks and for visualizing the task conditions to complement the syntactic-semantical sentence parsing. Two main difficulties in understanding texts of plane geometry tasks are observed: the ellipticity and vagueness of texts. To overcome the difficulties in understanding the task conditions it is proposed constructing cognitive models of objects and relations between them. The proposed cognitive approach is incorporated in an integrated system for automatic solving planimetric tasks with the natural language interface. The interactive visualization has been developed in the system. It depicts the syntactic and semantic structures as a result of natural language text analysis and searching for task solution. This visualization allows the users to obtain explanations associated with any elements of the images and to correct the tasks' texts in dialog with the system. The destiny of the system is to serve for training schoolchildren in the domain of Euclidean geometry. The cognitive approach proposed can be a first step to automated analyzing plane geometry texts, in perspective, as a cognitively controlled parsing. [ABSTRACT FROM AUTHOR]

We survey a few trace theorems for Sobolev spaces on N-dimensional Euclidean domains. We include known results on linear subspaces, in particular hyperspaces, and smooth boundaries, as well as less known results for Lipschitz boundaries, including Besov's Theorem and other characterizations of traces on planar domains, polygons in particular, in the spirit of the work of P. Grisvard. Finally, we present a recent approach, originally developed by G. Auchmuty in the case of the Sobolev space H1(Ω) on a Lipschitz domain Ω, and which we have further developed for the trace spaces of Hk(Ω), k ⩾ 2, by using Fourier expansions associated with the eigenfunctions of new multi-parameter polyharmonic Steklov problems. [ABSTRACT FROM AUTHOR]

We establish necessary and sufficient conditions for the compactness of fractional integral operators from Lp(X, μ) to Lq(X, μ) with 1 < p < q < ∞, where μ is a measure on a quasi-metric measure space X. As an application we obtain criteria for the compactness of fractional integral operators defined in weighted Lebesgue spaces over bounded domains of the Euclidean space ℝn with the Lebesgue measure, and also for the fractional integral operator associated to rectifiable curves of the complex plane. [ABSTRACT FROM AUTHOR]

We show that, for disjoint domains in the Euclidean space, mutual absolute continuity of their harmonic measures implies absolute continuity with respect to surface measure and rectifiability in the intersection of their boundaries. This improves on our previous result which assumed that the boundaries satisfied the capacity density condition. [ABSTRACT FROM AUTHOR]

We study bounded pseudoconvex domains in complex Euclidean space. We define an index associated to the boundary and show this new index is equivalent to the Diederich–Fornæss index defined in 1977. This connects the Diederich–Fornæss index to boundary conditions and refines the Levi pseudoconvexity. We also prove the β -worm domain is of index π / (2 β). It is the first time that a precise non-trivial Diederich–Fornæss index in Euclidean spaces is obtained. This finding also indicates that the Diederich–Fornæss index is a continuum in (0 , 1 ] , not a discrete set. The ideas of proof involve a new complex geometric analytic technique on the boundary and detailed estimates on differential equations. [ABSTRACT FROM AUTHOR]

We present novel families of wavelets and associated filterbanks for the analysis and representation of functions defined on circulant graphs. In this work, we leverage the inherent vanishing moment property of the circulant graph Laplacian operator, and by extension, the e-graph Laplacian, which is established as a parameterization of the former with respect to the degree per node, for the design of vertex-localized and critically-sampled higher-order graph (e-)spline wavelet filterbanks, which can reproduce and annihilate classes of (exponential) polynomial signals on circulant graphs. In addition, we discuss similarities and analogies of the detected properties and resulting constructions with splines and spline wavelets in the Euclidean domain. Ultimately, we consider generalizations to arbitrary graphs in the form of graph approximations, with focus on graph product decompositions. In particular, we proceed to show how the use of graph products facilitates a multi-dimensional extension of the proposed constructions and properties. [ABSTRACT FROM AUTHOR]

We establish an embedding theorem for spaces of functions of positive smoothness defined on irregular domains of n-dimensional Euclidean space in spaces of the same type and obtain some closely related results. [ABSTRACT FROM AUTHOR]

We consider the problem of estimating the volume of a compact domain in a Euclidean space based on a uniform sample from the domain. We assume that the domain has a boundary with positive reach. We propose a data-splitting approach to correct the bias of the plug-in estimator based on the sample α-convex hull. We show that this simple estimator achieves a minimax lower bound that we derive. Some numerical experiments corroborate our theoretical findings. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

We investigate sharp Trudinger-Moser type inequalities with the homogeneous weight satisfying a natural curvature-dimension bound condition. Also we study the optimal versions of these inequalities with best constants on both finite and infinite volume domains on Euclidean spaces. [ABSTRACT FROM AUTHOR]

Abstract We find necessary and sufficient conditions on a pullback diagram in order that every nonzero nonunit in its pullback ring admits a finite factorization into irreducible elements. As a result, we can describe a method of easily producing atomic domains that do not satisfy the ascending chain condition on principal ideals. [ABSTRACT FROM AUTHOR]