One-dimensional and two-dimensional continua belong to the basic notions of settheoretical topology and represent a subfield of the theory of dimensions developed by P. Urysohn and K. Menger. In this paper basic definitions and properties of grid continua in R2 and R3 are summarised. Particularly, simple one-dimensional grid continua in R2 and in R3, and simple closed two-dimensional grid conti...

Developing an adaptive 3-dimensional (3D) topology control algorithm is important because most wireless nodes are mobile and deployed in buildings. Moreover, in buildings, wireless link qualities and topologies change frequently due to various objects and the interference from other wireless devices. Previous topology control algorithms can suffer significant performance degradation because the...

In this paper, topology optimization is utilized for damage detection in three dimensional elasticity problems. In addition, two mode expansion techniques are used to derive unknown modal data from measured data identified by installed sensors. Damages in the model are assumed as reduction of mass and stiffness in the discretized finite elements. The Solid Isotropic Material with Penalization (...

Topology construction and topology maintenance are significant sub-problems of topology control. Spanning tree based algorithms for topology control are basically transmission range based type construction algorithms. The construction of an effective backbone, however, is indirectly related to the placement of nodes. Also, the dependence of network reliability on the communication path undertak...

Wireless Sensor Networks (WSNs) have been attracting the attention of both academic and industrial research communities for the past few years. To find an optimal topology control strategy in 3-dimensional (3D) WSNs is a new challenge. In this paper, we discuss issues related to the topology control in 3D WSNs, review a 2D model for WSNs, and introduce a 3D model for WSNs. An Immunity-Based Ant...

The early 1980’s saw enormous progress in understanding 4-manifolds: the topological Poincaré and annulus conjectures were proved, many cases of surgery and the s-cobordism theorem were settled, and Donaldson’s work showed that smooth structures are stranger than anyone had imagined. Big gaps remained: topological surgery and s-cobordisms with arbitrary fundamental group, and general classifica...

We present results concerning geometric topology of 2-dimensional polyhedra, with emphasis on those related to the problem of embeddability into 3-manifolds (and thickenings), the problem of resolving arbitrary 2-polyhedra by fake surfaces (including an application to reduce the classical Whitehead asphericity conjecture to special polyhedra) and existence of nonhomeomorphic 3-manifolds with eq...

Four-dimensional topology is in an unsettled state: a great deal is known, but the largest-scale patterns and basic unifying themes are not yet clear. Kirby has recently completed a massive review of low-dimensional problems [Kirby], and many of the results assembled there are complicated and incomplete. In this paper the focus is on a shorter list of \tool" questions, whose solution could unif...

A t the core of low-dimensional topology has been the classification of knots and links in the 3-sphere and the classification of 3and 4-dimensional manifolds (see Wikipedia for the definitions of basic topological terms). Beginning with the introduction of hyperbolic geometry into knots and 3-manifolds by W. Thurston in the late 1970s, geometric tools have become vital to the subject. Next cam...

SU(3) gauge theory in the 2-index symmetric (sextet) and fundamental representations is considered in symmetric and periodic boxes. Using the overlap formulation in the quenched approximation it is shown that the topological charge obtained from the sextet index theorem always leads to an integer value and agrees with the charge obtained from the fundamental index theorem in the continuum. At l...