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In this paper we consider reduction of nontorsion elements in the étale and Quillen K-theory of a curve X over a number field. As an application we solve two problems: detecting linear dependence and the support problem.

Settlement centers of various types, including cities, produce basins of attraction whose shape can be regular or complexly irregular (from the point of view of geometry). This complexity depends in part on properties of the space surrounding a settlement. This paper demonstrates that by introducing a dynamic approach to space and by including an equation of motion and space resistance, a dramatic change in the stylized static CPT (Central Place Theory) image occurs. As a result of the interplay of gravitational forces, basins of attraction arise around cities, whose boundaries appear to be fractals. This study provides a wealth of spatial fractal complex images which may change the traditional understanding of CPT.

In this paper we consider divisibility sequences obtained from square matrices. We work with of matrix divisibility sequences associated to a semigroup and arising from endomorphisms of an affine space. We prove that determinant divisibility sequences originated from powers of square matrices are generalized Lucas sequences.

Settlement centers of various types, including cities, produce basins of attraction whose shape can be regular or complexly irregular (from the point of view of geometry). It depends, among others, on the properties of the space surrounding a city. Central Place Theory (CPT), a well-known concept in economic geography and spatial economics, explains the spatial structure inherent in distinct economic orders of urban centers in geographical space. In Christaller’s original formulation the fundamental architecture of CPT is based on a static approach which focuses on the optimal division of space around urban centers, while following pre-specified plausible rules of supplying them with goods from other places. Conducted in the spirit of recent works, this study demonstrates that by introducing a dynamic approach to space and including an equation of motion and space resistance, a dramatic change of the idealized static CPT image occurs. As a result of the interplay of gravitational forces, basins of attraction arise around cities, whose boundaries are fractals. This paper provides a wealth of spatial fractal complex images that question the relevance and validity of traditional CPT.