Hyperspherical Variational Co-embedding for Attributed Networks

Network-based information has been widely explored and exploited in the information retrieval literature. Attributed networks, consisting of nodes, edges as well as attributes describing properties of nodes, are a basic type of network-based data, and are especially useful for many applications. Examples include user profiling in social networks and item recommendation in user-item purchase networks. Learning useful and expressive representations of entities in attributed networks can provide more effective building blocks to down-stream network-based tasks such as link prediction and attribute inference. Practically, input features of attributed networks are normalized as unit directional vectors. However, most network embedding techniques ignore the spherical nature of inputs and focus on learning representations in a Gaussian or Euclidean space, which, we hypothesize, might lead to less effective representations. To obtain more effective representations of attributed networks, we investigate the problem of mapping an attributed network with unit normalized directional features into a non-Gaussian and non-Euclidean space. Specifically, we propose a hyperspherical variational co-embedding for attributed networks (HCAN), which is based on generalized variational auto-encoders for heterogeneous data with multiple types of entities. HCAN jointly learns latent embeddings for both nodes and attributes in a unified hyperspherical space such that the affinities between nodes and attributes can be captured effectively. We argue that this is a crucial feature in many real-world applications of attributed networks. Previous Gaussian network embedding algorithms break the assumption of uninformative prior, which leads to unstable results and poor performance. In contrast, HCAN embeds nodes and attributes as von Mises-Fisher distributions, and allows one to capture the uncertainty of the inferred representations. Experimental results on eight datasets show that HCAN yields better performance in a number of applications compared with nine state-of-the-art baselines.

(ζ−m, ζm)-Type Algebraic Minimal Surfaces in Three-Dimensional Euclidean Space

We introduce the real minimal surfaces family by using the Weierstrass data (ζ−m,ζm) for ζ∈C, m∈Z≥2, then compute the irreducible algebraic surfaces of the surfaces family in three-dimensional Euclidean space E3. In addition, we propose that family has a degree number (resp., class number) 2m(m+1) in the cartesian coordinates x,y,z (resp., in the inhomogeneous tangential coordinates a,b,c).

Pure rolling motion of hyperquadrics in pseudo-Euclidean spaces

<p style='text-indent:20px;'>This paper is devoted to rolling motions of one manifold over another of equal dimension, subject to the nonholonomic constraints of no-slip and no-twist, assuming that these motions occur inside a pseudo-Euclidean space. We first introduce a definition of rolling map adjusted to this situation, which generalizes the classical definition of Sharpe [<xref ref-type="bibr" rid="b26">26</xref>] for submanifolds of an Euclidean space. We also prove some important properties of these rolling maps. After presenting the general framework, we analyse the particular rolling of hyperquadrics embedded in pseudo-Euclidean spaces. The central topic is the rolling of a pseudo-hyperbolic space over the affine space associated with its tangent space at a point. We derive the kinematic equations, as well as the corresponding explicit solutions for two specific cases, and prove the existence of a rolling map along any curve in that rolling space. Rolling of a pseudo-hyperbolic space on another and rolling of pseudo-spheres are equally treated. Finally, for the central theme, we write the kinematic equations as a control system evolving on a certain Lie group and prove its controllability. The choice of the controls corresponds to the choice of a rolling curve.</p>

On parallel p-equidistant ruled surfaces by using mofied orthogonal frame with curvature in E³

In this paper, it is investigated Ruled surfaces according to modified orthogonal frame with curvature in 3-dimensional Euclidean space. Firstly, we give apex angle, pitch and drall of closed ruled surface in E³. Then, it characterized the relationship between these invariant of parallel p-equidistant ruled surfaces.

DIFFERENTIAL GAME WITH A LIFELINE FOR THE INERTIAL MOVEMENTS OF PLAYERS

In this paper, we study the well-known problem of Isaacs called the "Life line" game when movements of players occur by acceleration vectors, that is, by inertia in Euclidean space. To solve this problem, we investigate the dynamics of the attainability domain of an evader through finding solvability conditions of the pursuit-evasion problems in favor of a pursuer or an evader. Here a pursuit problem is solved by a parallel pursuit strategy. To solve an evasion problem, we propose a strategy for the evader and show that the evasion is possible from given initial positions of players. Note that this work develops and continues studies of Isaacs, Petrosjan, Pshenichnii, Azamov, and others performed for the case of players' movements without inertia.

Surfaces of Finite III-type in the Euclidean 3-Space

In this paper, we firstly investigate some relations regarding the first and the second Laplace operators corresponding to the third fundamental form III of a surface in the Euclidean space E3. Then, we introduce the finite Chen type surfaces of revolution with respect to the third fundamental form which Gauss curvature never vanishes.