Euclidean Space Research Articles

Pore‐throat structure and fractal characteristics of tight gas sandstone reservoirs: A case study of the second member of the Upper Triassic Xujiahe Formation in Zhongba area, western Sichuan depression, China

Pore‐throat structure is a key factor that influences the storage and fluid flow capacity of tight sandstone reservoirs. Taking the tight sandstone reservoir of Xu2 Fm in Zhongba area as an example, the reservoir quality, pore‐throat type and pore size distribution of tight sandstone in the study area were described by casting thin section, scanning electron microscope and high‐pressure mercury injection test. In order to quantitatively characterize the complexity and heterogeneity of pore‐throat structure, fractal analysis was performed using mercury saturation and pore size data. This study mainly reveals the relationship between the geometric shape characteristics and fractal dimension of the binary pore structure of ultra‐low permeability tight sandstone and clarifies the influence of different scale pore throats on reservoir physical properties. The results indicate that the physical properties of the reservoir in the study area are poor, the pores are mainly intergranular pores and dissolution pores, the throat is flake and necked and the pore size distribution range is large. The fractal curve obtained by the mercury saturation method shows a significant turning point, and the pore‐throat system is divided into two types: small‐scale and large‐scale. The fractal dimension of large‐scale pore throat is greater than the three‐dimensional Euclidean space dimension, which does not conform to the fractal theory. The fractal dimension of small‐scale pore throat is closely related to the pore‐throat structure and has obvious fractal characteristics. The geometric shape of binary pore‐throat structure in tight sandstone is the main factor affecting the difference of fractal dimension. The development of small pores in sandstone is positively correlated with the total porosity, but its contribution to permeability is relatively low. The physical properties of tight sandstone are mainly controlled by the development degree of large‐scale pore throat.

Sparse Signal Recovery via Rescaled Matching Pursuit

We propose the Rescaled Matching Pursuit (RMP) algorithm to recover sparse signals in high-dimensional Euclidean spaces. The RMP algorithm has less computational complexity than other greedy-type algorithms, such as Orthogonal Matching Pursuit (OMP). We show that if the restricted isometry property is satisfied, then the upper bound of the error between the original signal and its approximation can be derived. Furthermore, we prove that the RMP algorithm can find the correct support of sparse signals from random measurements with a high probability. Our numerical experiments also verify this conclusion and show that RMP is stable with the noise. So, the RMP algorithm is a suitable method for recovering sparse signals.

Imaging n-Dimensional Spaces Within m-Dimensional Spaces: An Extension of Hinton’s Method

We derive a Euclidean manifold that is virtually cyclic using a simple equation based on a Euclidean geometry and related to Hinton’s method. The derived equation is simple to understand, but able to project n-dimensional spaces into m-dimensional spaces. In addition, the method produces exact images of rectangular cuboids as elements of a vector space, implying that information is a vector and not a scalar.

Extending Natural Mates in Euclidean 3-space and Applications to Bertrand Pairs

In Euclidean 3-space, a family of curves, the co-successor, is motivated and then introduced in relation to the natural mate. A complete characterization of co-successors is proved, followed by an application of the co-successor towards describing Bertrand curves and their mates.

Rotational Self-Shrinkers in Euclidean Spaces

The rotational embedded submanifold of $\mathbb{E}^{n+d}$ first studied by N. Kuiper. The special examples of this type are generalized Beltrami submanifolds and toroidals submanifold. The second named authour and at. all recently have considered $3-$dimensional rotational embedded submanifolds in $\mathbb{E}^{5}$. They gave some basic curvature properties of this type of submaifolds. Self-similar flows emerge as a special solution to the mean curvature flow that preserves the shape of the evolving submanifold. In this article we consider self-similar submanifolds in Euclidean spaces. We obtained some results related with self-shrinking rotational submanifolds in Euclidean $5-$space $\mathbb{E}^{5}$. Moreover, we give the necessary and sufficient conditions for these type of submanifolds to be homothetic solitons for their mean curvature flows.

Construction of Ruled Surfaces from the W-Curves and Their Characterizations in E3

Ruled surfaces are considered one of the significant aspects of differential geometry. These surfaces are formed by the motion of a straight line called a generator, and every curve that intersects all the generators is called a directrix. In the present research paper, we explore a family of ruled surfaces constructed from circular helices (W-curve) using the Frenet frame in the Euclidean space E3. We derive the explicit formulas for the second mean curvature and second Gaussian curvature. We present some ruled surfaces, and we describe their properties. In addition, we determine the sufficient conditions for these surfaces to be minimal, flat, II-minimal, and II-flat. Also, we obtain sufficient conditions for the base curve for these ruled surfaces to be a geodesic curve, an asymptotic line, and a principal line. Furthermore, we present an application for a ruled surface whose base curve is a circular helix, we compute some quantities for this surface such as the mean curvature and Gaussian curvatures and we plot the ruled surface with its base curve, and at symmetric points and along a symmetry axis.

Rotational Surfaces in $\mathbb R^4$ with New Approaches and Examples

In this paper, we give a new approach to the rotational minimal surfaces in $4$-dimensional Euclidean space $\mathbb{R}^4$. One type of these surfaces is obtained by the composition of two families of rotations in orthogonal planes. For these surfaces, we give a new parameterization. Using this parametrization, we find new examples of rotational minimal surfaces and rotational surfaces with zero Gaussian curvature.

New Parametrization of Bertrand Partner D-curves in $\mathbb{E}^{3}$

We define and study a new parametrization of a Bertrand partner D-curve \{$\alpha$,$\alpha^{*}$\} in Euclidean 3-space by not taking Darboux frame element $g^{*}$ of Bertrand partner D-curve $\alpha^{*}$ parallel to $\overrightarrow{\alpha \alpha^{*}}$. We obtain a necessary and sufficient condition for a curve to be such type of Bertrand D-curves. Also, we obtain a characterization of a new parametrization of asymptotic Bertrand D-curves and provide an example.

Hyperbolic multichannel graph convolutional network for social recommendation

Social recommendations enhance the quality of recommendations by integrating social network information. Existing methods predominantly rely on pairwise relationships to uncover potential user preferences. However, they usually overlook the exploration of higher-order user relations. Moreover, because social relation graphs often exhibit scale-free graph structures, directly embedding them in Euclidean space will lead to significant distortion. To this end, we propose a novel graph neural network framework with hypergraph and hyperbolic embedding learning, namely HMGCN. Specifically, we first construct hypergraphs over user-item interactions and social networks, and then perform graph convolution on the hypergraphs. At the same time, a multi-channel setting is employed in the convolutional network, with each channel encoding its corresponding hypergraph to capture different high-order user relation patterns. In addition, we feed the item embeddings and the obtained high-order user embeddings into a hyperbolic graph convolutional network to extract user and item representations, enabling the model to better capture the hierarchical structure of their complex relationships. Experimental results on three public datasets, namely FilmTrust, LastFM, and Yelp, demonstrate that the model achieves more comprehensive user and item representations, more accurate fitting and processing of graph data, and effectively addresses the issues of insufficient user relationship extraction and data embedding distortion in social recommendation models.

Wormhole solutions in [formula omitted] gravity generated by new shape functions and their complexity factor

The objective of this paper is to investigate the existence and stability of wormholes within the framework of modified gravity, specifically F(R,Lm) gravity, where F(R) represents a function of the Ricci scalar and Lm accounts for matter Lagrangian. We consider the linear functional form of F(R,Lm) with two novel shape functions. To illustrate the wormhole configurations, the embedding diagram in two and three-dimensional Euclidean space is also examined. The Tolman–Oppenheimer–Volkoff (TOV) equation is also used to investigate the stability of the created framework of the traversable wormholes. We observed that near the wormhole throat, the complexity factor (ycf) is monotonically decreasing for wormhole Model-I and increasing for wormhole Model-II, and for higher values of the radial coordinate, ycf is getting closer to zero. Our analysis shows that the energy conditions have been violated and that this generic violation points to the presence of exotic matter.

Stability estimates for an inverse boundary value problem for biharmonic operators with first order perturbation from partial data

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.

Existence and asymptotic behavior for L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-norm preserving nonlinear heat equations

We consider a nonlinear parabolic equation with a nonlocal term which preserves the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-norm of the solution. We study the local and global well-posedness on a bounded domain, as well as the whole Euclidean space, in H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1$$\\end{document}. Then we study the asymptotic behavior of solutions. In general, we obtain weak convergence in H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1$$\\end{document} to a stationary state. For a ball, we prove strong convergence to the ground state when the initial condition is positive.

Random vector functional link networks for function approximation on manifolds

The learning speed of feed-forward neural networks is notoriously slow and has presented a bottleneck in deep learning applications for several decades. For instance, gradient-based learning algorithms, which are used extensively to train neural networks, tend to work slowly when all of the network parameters must be iteratively tuned. To counter this, both researchers and practitioners have tried introducing randomness to reduce the learning requirement. Based on the original construction of Igelnik and Pao, single layer neural-networks with random input-to-hidden layer weights and biases have seen success in practice, but the necessary theoretical justification is lacking. In this study, we begin to fill this theoretical gap. We then extend this result to the non-asymptotic setting using a concentration inequality for Monte-Carlo integral approximations. We provide a (corrected) rigorous proof that the Igelnik and Pao construction is a universal approximator for continuous functions on compact domains, with approximation error squared decaying asymptotically like O(1/n) for the number n of network nodes. We then extend this result to the non-asymptotic setting, proving that one can achieve any desired approximation error with high probability provided n is sufficiently large. We further adapt this randomized neural network architecture to approximate functions on smooth, compact submanifolds of Euclidean space, providing theoretical guarantees in both the asymptotic and non-asymptotic forms. Finally, we illustrate our results on manifolds with numerical experiments.

Solving a class of multi-scale elliptic PDEs by Fourier-based mixed physics informed neural networks

Deep neural networks have garnered widespread attention due to their simplicity and flexibility in the fields of engineering and scientific calculation. In this study, we probe into solving a class of elliptic partial differential equations (PDEs) with multiple scales by utilizing Fourier-based mixed physics informed neural networks (dubbed FMPINN), its solver is configured as a multi-scale deep neural network. In contrast to the classical PINN method, a dual (flux) variable about the rough coefficient of PDEs is introduced to avoid the ill-condition of neural tangent kernel matrix caused by the oscillating coefficient of multi-scale PDEs. Therefore, apart from the physical conservation laws, the discrepancy between the auxiliary variables and the gradients of multi-scale coefficients is incorporated into the cost function, obtaining a satisfactory solution of PDEs by minimizing the defined loss through some optimization methods. Additionally, a trigonometric activation function is introduced for FMPINN, which is suited for representing the derivatives of complex target functions. Handling the input data by Fourier feature mapping will effectively improve the capacity of deep neural networks to solve high-frequency problems. Finally, to validate the efficiency and robustness of the proposed FMPINN algorithm, we present several numerical examples of multi-scale problems in various dimensional Euclidean spaces. These examples cover low-frequency and high-frequency oscillation cases, demonstrating the effectiveness of our approach. All code and data accompanying this manuscript will be publicly available at https://github.com/Blue-Giant/FMPINN.

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A unified Fourier slice method to derive ridgelet transform for a variety of depth-2 neural networks

To investigate neural network parameters, it is easier to study the distribution of parameters than to study the parameters in each neuron. The ridgelet transform is a pseudo-inverse operator that maps a given function f to the parameter distribution γ so that a network NN[γ] reproduces f, i.e. NN[γ]=f. For depth-2 fully-connected networks on a Euclidean space, the ridgelet transform has been discovered up to the closed-form expression, thus we could describe how the parameters are distributed. However, for a variety of modern neural network architectures, the closed-form expression has not been known. In this paper, we explain a systematic method using Fourier expressions to derive ridgelet transforms for a variety of modern networks such as networks on finite fields Fp, group convolutional networks on abstract Hilbert space H, fully-connected networks on noncompact symmetric spaces G/K, and pooling layers, or the d-plane ridgelet transform.

Manifold-based multi-graph embedding for semi-supervised classification

Graph semi-supervised learning (GSSL) plays an important role in semi-supervised classification by leveraging the similarity of graph topology and convex optimization with Laplacian-based regularization. Current approaches mainly focus on data representation in the Euclidean space and solely rely on a single graph structure, ignoring the underlined manifold-valued structure of the data. To this end, we propose a novel manifold-based multi-graph embedding (M2GE), which explicitly learns representations on both Riemannian manifold and Euclidean space in the embedding learning process. Incorporating multi-graph structure, the rich and varied features are used by M2GE to more fully capture categorical information, where the heterogeneous spatial representations from manifold are complementary. Extensive experimental results on three popular benchmarks verify demonstrate the superiority of our method over state-of-the art competitors.

NSimplex Zen : A Novel Dimensionality Reduction for Euclidean and Hilbert Spaces

Dimensionality reduction techniques map values from a high dimensional space to one with a lower dimension. The result is a space which requires less physical memory and has a faster distance calculation. These techniques are widely used where required properties of the reduced-dimension space give an acceptable accuracy with respect to the original space. Many such transforms have been described. They have been classified in two main groups: linear and topological . Linear methods such as Principal Component Analysis (PCA) and Random Projection (RP) define matrix-based transforms into a lower dimension of Euclidean space. Topological methods such as Multidimensional Scaling (MDS) attempt to preserve higher-level aspects such as the nearest-neighbour relation, and some may be applied to non-Euclidean spaces. Here, we introduce nSimplex Zen , a novel topological method of reducing dimensionality. Like MDS, it relies only upon pairwise distances measured in the original space. The use of distances, rather than coordinates, allows the technique to be applied to both Euclidean and other Hilbert spaces, including those governed by Cosine, Jensen–Shannon and Quadratic Form distances. We show that in almost all cases, due to geometric properties of high-dimensional spaces, our new technique gives better properties than others, especially with reduction to very low dimensions.

A note on umbilic points at infinity

AbstractIn this note a definition of umbilic point at infinity is proposed, at least for surfaces that are homogeneous polynomial graphs over a plane in Euclidean 3-space. This is a stronger definition than that of Toponogov in his study of complete convex surfaces, and allows one to distinguish between different umbilic points at infinity. It is proven that all such umbilic points at infinity are isolated, that they occur in pairs and are the zeroes of the projective extension of the third fundamental form, as developed in Guilfoyle and Ortiz-Rodríguez (Math Proc R Ir Acad 123A(2), 63–94, 2023). A geometric interpretation for the definition is that an umbilic point at infinity occurs when the tangent to the level set at infinity is also an asymptotic direction at infinity.

Characterization of Tubular Surfaces in Terms of Finite III-type

In this paper, we first define relations regarding the first and the second Laplace operators corresponding to the third fundamental form III of a surface in the Euclidean space E 3 . Then, we will characterize the tubular surfaces in terms of their coordinate finite type.