Narrow Results

This work is devoted to interpretation of concepts of Zariski dimension of an algebraic variety over a field and of Krull dimension of a coordinate ring in algebraic geometry over algebraic structures of an arbitrary signature. Proposed dimensions are ordinal numbers (ordinals).

Artificial Intelligence focuses on the question of how to design system to exhibit intelligent behaviour in complex environments. Complex global behaviours can emerge from simple systems acting in a complex environment; however, this emergence requires that the systems' internal structure reflect essential structures in the environment. This paper examines the algebraic structure of a system's actions. We find that these actions often possess a self-similar local neighborhood structure that permits analysis and synthesis to be performed locally yet produce global, intelligent behaviours. A procedure for finding this local structure is presented, and illustrated with examples.

Clustering, as one of the main tasks of machine learning, is also the core work of granular computing, namely granulation. Most of the recent granular computing based clustering algorithms only utilize the plain granule features without taking the granule structure into account, especially in information area with widespread application of algebraic structure. This paper aims at proposing an algebraic structure based clustering method from granular computing prospective. Specifically, the algebraic structure based granularity is firstly formulated based on the granule structure of an algebraic binary operator. An algebraic structure based clustering method is then proposed by incorporating congruence partitioning granules and homomorphically projecting granule structure. Finally, proof of the lattice at multiple hierarchical levels and comparative analysis of experimental cases validate the effectiveness of the proposed clustering method. The algebraic structure based clustering method can provide a general framework to perform granularity clustering using the algebraic granule structure information. It meanwhile advances the granular computing methods by combing the granular computing theory and the clustering theory.

Physics at very high energies (close to the Planck energy scale) is in the center of modern physics challenges. At the high energy scales, or equivalently very short distances, the very notion of spacetime acquires a complicated structure. In this scale, there are limitations on the complete resolution of adjacent points in spacetime manifold. While the effect of a minimal measurable length on algebraic structure of a free particle’s phase space has been studied in the literature, the simultaneous effects of minimal length and maximal momentum have not been explored in this subject so far. Here, we are going to fill this gap. We study some properties of the algebraic structure of quantum field theories in the presence of natural cutoffs as a minimal length and a maximal momentum in a free particle’s phase space. Especially, we pay attention on the phase space trajectories of systems in the presence of natural cutoffs.

In this paper, we have established the concept of mutant soft set for algebraic formalization of biological mutation, and examined its structural properties.