Closed, oriented, connected 3-manifolds are subtle equivalence classes of plane graphs

A blink is a plane graph with an arbitrary bipartition of its edges. As a consequence of a recent result of Martelli, it is shown that the homeomorphisms classes of closed oriented 3-manifolds are in 1-1 correspondence with specific classes of blinks. In these classes, two blinks are equivalent if they are linked by a finite sequence of local moves, where each one appears in a concrete list of 64 moves: they are organized in 8 types, each being essentially the same move on 8 simply related configurations. The size of the list can be substantially decreased at the cost of loosing symmetry, just by keeping a very simple move type, the ribbon moves denoted [Formula: see text] (which are in principle redundant). The inclusion of [Formula: see text] implies that all the moves corresponding to plane duality (the starred moves), except for [Formula: see text] and [Formula: see text], are redundant and the coin calculus is reduced to 36 moves on 36 coins. A residual fraction link or a flink is a new object which generalizes blackboard-framed link. It plays an important role in this work. It is in the aegis of this work to find new important connections between 3-manifolds and plane graphs.

ON mRJ REACHABILITY IN TREES

Given a tree T, a configuration of T is denoted by [Formula: see text] which represents that there is a robot at the vertex u, a hole at the vertex v and obstacles in the remaining vertices of T. By an mRJ move we mean that the robot is moved from the vertex u to a vertex v having a hole by jumping over m obstacles along a path. The case m = 0 is a simple move of taking the robot from u to the adjacent vertex v with a hole. We investigate the problem of moving a robot from its initial position to all the other vertices using mRJ moves (for some fixed m) in addition to simple moves. A tree is said to be mRJ reachable if there exists a configuration from which it is possible to take the robot to any vertex of the tree using simple or mRJ moves. A connected graph is 1RJ reachable. However, for m ≥ 2 there exists graphs that are not mRJ reachable. We characterize 2RJ and 3RJ reachable trees and give bound for the diameter of mRJ reachable trees.

Influence of Shell Geometry on Buckling Optimization of Fiber-Composite Laminate Shells

The buckling strength of fiber-composite laminate shells with given loading condition and material system is maximized with respect to fiber orientations by using a sequential linear programming method together with a simple move-limit strategy. Significant influence of shell thickness and shell length on the optimal fiber angles, the critical buckling loads and the critical buckling modes of fiber-composite laminate shells has been shown through this investigation.

Ante Ciliga, Trotskii, and State Capitalism: Theory, Tactics, and Reevaluation during the Purge Era, 1935-1939

The Moscow Trials forced the international anti-Stalinist Left to reconsider its most fundamental truths. The old, rigid categories of capitalism and socialism were no longer certain; a new urgency accompanied the question of where the revolution had gone astray. Many of the most outspoken anti-Stalinists searching for new answers had been either adherents or sympathizers of Trotskii. For many of these former staunch supporters, Trotskii's theoretical middle ground—at the same time hostile to the Stalinist bureaucracy but defensive of the land of socialized production— was no longer tenable. One after another, they broke with Trotskii. Yet this entire movement of reevaluation in Trotskii's intellectual entourage either has not been fully explored in discussions centering on Trotskii's views or, in a certain Trotskyist tradition, has been dismissed as a simple move to the Right. One of the best ways to understand both the roots and the context of such an intellectual upheaval is to focus on the key individuals.