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Solving R. J. Daverman’s problem, V. S. Krushkal described sticky Cantor sets in ${ℝ}^N$ for $N\ge 4$. Such sets cannot be isotoped off themselves by small ambient isotopies. Using Krushkal sets, we present a new series of wild embeddings related to a question of J. W. Cannon and S. G. Wayment (1970). Namely, for $N\ge 4$, we construct examples of compacta $X\subset {ℝ}^N$ with the following two properties: some sequence $\{X_i\subset {ℝ}^N\setminus X,i\in \mathbb{N}\}$ converges homeomorphically to $X$, but no uncountable family of pairwise disjoint sets $Y_{\alpha }\subset {ℝ}^N$ exists such that each $Y_{\alpha }$ is ambiently homeomorphic to $X$.

In the present paper we classify curves and surfaces in Euclidean 3-space which make constant angle with a certain Killing vector field. Moreover, we characterize the catenoid and Dini's surface in terms of constant angle surfaces.

Many real-world networks are essentially heterogeneous, where the nodes have different abilities to gain connections. Such networks are difficult to be embedded into low-dimensional Euclidean space if we ignore the heterogeneity and treat all the nodes equally. In this paper, based on a newly defined heterogeneous distance and a generalized network distance under the constraints of network and triangle inequalities, respectively, we propose a new heterogeneous multidimensional scaling method (HMDS) to embed different networks into proper Euclidean spaces. We find that HMDS behaves much better than the traditional multidimensional scaling method (MDS) in embedding different artificial and real-world networks into Euclidean spaces. Besides, we also propose a method to estimate the appropriate dimensions of Euclidean spaces for different networks, and find that the estimated dimensions are quite close to the real dimensions for those geometrical networks under study. These methods thus can help to better understand the evolution of real-world networks, and have practical importance in network visualization, community detection, link prediction and localization of wireless sensors.

V. V. Grushin and V. P. Palamodov proved in 1962 that it is impossible to place in $R^3$ uncountably many pairwise disjoint polyhedra each homeomorphic to the Moebius band. We generalize this result in two directions. First, we give a generalization of this result to tame subsets in $R^N$, $N\ge 3$. Second, we show that in case of $R^3$ the theorem holds for arbitrarily topologically embedded (not necessarily tame) Moebius bands.

A topoplane is a mild deformation of a linear hyperplane contained in a given smooth manifold that is homeomorphic to a Euclidean space. We consider solidly transsective topoplane arrangements. These collections generalize pseudohyperplane arrangements. Even though the topoplane arrangements locally look like hyperplane arrangements, the global coning procedure is absent here. The main aim of the paper is to introduce the Varchenko matrix in this context and show that the determinant has a similar factorization as in the case of hyperplane arrangements. We achieve this by suitably generalizing the strategy of Aguiar and Mahajan. We also study a system of linear equations introduced by them and describe its solution space in the context of topoplane arrangements.

A root system in which all roots have same norm is known as a simply laced root system. We present a simple method of classifying all simply laced root systems.

In this work, we investigate relationships between Darboux and type-2 Bishop frames in Euclidean space. Then, we obtain the geodesic curvature of the spherical image curve of the Darboux vector of the type-2 Bishop frame. Also, we give transition matrix between the Darboux and type-2 Bishop frames of the type-2 Bishop frames of the spherical images of the edges ${\overrightarrow{N}}_1,{\overrightarrow{N}}_2$ and $\overrightarrow{b}$. Finally, we express some interesting relations and illustrate of the examples by the aid Maple programe.

There are two smooth functions $\sigma$ and $\rho$ associated to a nontrivial concircular vector field $v$ on a connected Riemannian manifold $(M,g)$, called potential function and connecting function. In this paper, we show that presence of a timelike nontrivial concircular vector field influences the geometry of generalized Robertson–Walker space-times. We use a timelike concircular vector field $v$ on an $n$ -dimensional connected conformally flat Lorentzian manifold, $n>2$, to find a characterization of generalized Robertson–Walker space-time with fibers Einstein manifolds. It is interesting to note that for $n=4$ the concircular vector field annihilates energy-momentum tensor and also that in this case the potential function $\sigma$ is harmonic. In the second part of this paper, we show that presence of a nontrivial concircular vector field $v$ with connecting function $\rho$ on a complete and connected $n$ -dimensional conformally flat Riemannian manifold $(M,g)$, $n>2$, with Ricci curvature $\mathrm{\text{Ric}}(v,v)$ non-negative, satisfying $n(n-1)\rho +\tau =0$, is necessary and sufficient for $(M,g)$ to be isometric to either a sphere $S^n(c)$ or to the Euclidean space $E^n$, where $\tau$ is the scalar curvature.

We have posed a simple but interesting graph theoretic problem and posited a heuristic solution procedure, which we have christened as Vectored Route-lenght Minimization (VeRMin). Basically, it constitutes a re-casting of the classical "shortest route" problem in a strictly Euclidean space. We have presented only a heuristic solution process hoping that a formal proof will eventually emerge as the problem receives wider exposure within mathematical circles.

Let V be a finite dimensional complex vector space equipped with an inner product. Let G denote the group of all affine automorphisms of V preserving the metric defined by the inner product. Let H be a connected reductive affine algebraic group defined over ℂ. We give an explicit classification of the isomorphism classes of G-equivariant holomorphic hermitian principal H-bundles over V.

A Steiner minimal tree is a network with minimum length spanning a given set of points in space. There are several criteria for identifying the Steiner minimal tree on four points in the Euclidean plane. However, it has been proved that the length of the Steiner minimal tree on four points cannot be computed using radicals if the four points lie in Euclidean space. This unsolvability implies that it is unlikely that similar necessary and sufficient conditions exist in the spatial case. Hence, a problem arises: Is it possible to generalize the known planar criteria to space in the sense that they are sufficient to identify Steiner minimal trees on four points in space? This problem is investigated and some sufficient conditions are proved in this paper. These sufficient conditions can help us to solve the general Steiner tree problem on n(> 4) points in Euclidean space.

Combinatorial theorems of real convex and complex analysis are considered. We give some generalizations of classical Caratheodory and Helly theorems.