Computational complexity of diagram satisfaction in Euclidean geometry

AbstractIn this paper, it is shown that the problem of deciding whether or not a geometric diagram in Euclidean Geometry is satisfiable is NP-hard and in PSPACE, and in fact has the same complexity as the satisfaction problem for a fragment of the existential theory of the real numbers. The related problem of finding all of the possible (satisfiable) diagrams that can result when a segment of a diagram is extended is also shown to be NP-hard

Moving Walkways, Escalators, and Elevators

We study a simple geometric model of transportation facility that consists of two points between which the travel speed is high. This elementary definition can model shuttle services, tunnels, bridges, teleportation devices, escalators or moving walkways. The travel time between a pair of points is defined as a time distance, in such a way that a customer uses the transportation facility only if it is helpful. We give algorithms for finding the optimal location of such a transportation facility, where optimality is defined with respect to the maximum travel time between two points in a given set.Comment: 16 pages. Presented at XII Encuentros de Geometria Computacional, Valladolid, Spai

Theoretical Engineering and Satellite Comlink of a PTVD-SHAM System

This paper focuses on super helical memory system's design, 'Engineering, Architectural and Satellite Communications' as a theoretical approach of an invention-model to 'store time-data'. The current release entails three concepts: 1- an in-depth theoretical physics engineering of the chip including its, 2- architectural concept based on VLSI methods, and 3- the time-data versus data-time algorithm. The 'Parallel Time Varying & Data Super-helical Access Memory' (PTVD-SHAM), possesses a waterfall effect in its architecture dealing with the process of voltage output-switch into diverse logic and quantum states described as 'Boolean logic & image-logic', respectively. Quantum dot computational methods are explained by utilizing coiled carbon nanotubes (CCNTs) and CNT field effect transistors (CNFETs) in the chip's architecture. Quantum confinement, categorized quantum well substrate, and B-field flux involvements are discussed in theory. Multi-access of coherent sequences of 'qubit addressing' in any magnitude, gained as pre-defined, here e.g., the 'big O notation' asymptotically confined into singularity while possessing a magnitude of 'infinity' for the orientation of array displacement. Gaussian curvature of k(k<0) is debated in aim of specifying the 2D electron gas characteristics, data storage system for defining short and long time cycles for different CCNT diameters where space-time continuum is folded by chance for the particle. Precise pre/post data timing for, e.g., seismic waves before earthquake mantle-reach event occurrence, including time varying self-clocking devices in diverse geographic locations for radar systems is illustrated in the Subsections of the paper. The theoretical fabrication process, electromigration between chip's components is discussed as well.Comment: 50 pages, 10 figures (3 multi-figures), 2 tables. v.1: 1 postulate entailing hypothetical ideas, design and model on future technological advances of PTVD-SHAM. The results of the previous paper [arXiv:0707.1151v6], are extended in order to prove some introductory conjectures in theoretical engineering advanced to architectural analysi

Geometric Embeddability of Complexes Is ??-Complete

We show that the decision problem of determining whether a given (abstract simplicial) k-complex has a geometric embedding in ?^d is complete for the Existential Theory of the Reals for all d ? 3 and k ? {d-1,d}. Consequently, the problem is polynomial time equivalent to determining whether a polynomial equation system has a real solution and other important problems from various fields related to packing, Nash equilibria, minimum convex covers, the Art Gallery Problem, continuous constraint satisfaction problems, and training neural networks. Moreover, this implies NP-hardness and constitutes the first hardness result for the algorithmic problem of geometric embedding (abstract simplicial) complexes. This complements recent breakthroughs for the computational complexity of piece-wise linear embeddability