Mathematics

Geometric properties of the fixed point set Fix(f) of a self-mapping f on a metric or a generalized metric space is an attractive issue. The set Fix(f) can contain a geometric figure (a circle, an ellipse, etc.) or it can be a geometric figure. In this paper, we consider the set of simulation functions for geometric applications in the fixed point theory both on metric and some generalized metric spaces ( S -metric spaces and b -metric spaces). The main motivation of this paper is to investigate the geometric properties of non unique fixed points of self-mappings via simulation functions.

The first goal of this article is to provide an statement of the conditions for geometric continuity of order k, referred in the bibliography as beta-constraints, in terms of Riordan matrices. The second one is to see this new formulation in action to solve a theoretical cuestion about uniqueness of analytic solution for a general and classical problem in plane geometry: the F-chordal problem.

This paper is devoted to the study of tangential properties of measures with density in the Heisenberg groups H n . Among other results we prove that measures with (2n+1) -density have only flat tangents and conclude the classification of uniform measures in H 1 .

The Grassmann angle improves upon similar angles between subspaces that measure volume contraction in orthogonal projections. It works in real or complex spaces, with important differences, and is asymmetric, what makes it more efficient when dimensions are distinct. It can be seen as an angle in Grassmann algebra, being related to its products and those of Clifford algebra, and gives the Fubini-Study metric on Grassmannians, an asymmetric metric on the full Grassmannian, and Hausdorff distances between full sub-Grassmannians. We give formulas for computing it in arbitrary bases, and identities for angles with certain families of subspaces, some of which are linked to real and complex Pythagorean theorems for volumes and quantum probabilities. Unusual features of the angle with an orthogonal complement, or the angle in complex spaces, are examined.

For any integer n≥2 , a square can be partitioned into n 2 smaller squares via a checkerboard-type dissection. Does there such a class-preserving grid dissection exist for some other types of quadrilaterals? For instance, is it true that a tangential quadrilateral can be partitioned into n 2 smaller tangential quadrilaterals using an n×n grid dissection? We prove that the answer is affirmative for every integer n≥2 .

For a contractive iterated function system (IFS), it is known that there is a natural hyperbolic graph structure (augmented tree) on the symbolic space of the IFS that reflects the relationship among neighboring cells, and its hyperbolic boundary with the Gromov metric is Hölder equivalent to the attractor K . This setup was taken up to study the probabilistic potential theory on K , and the bi-Lipschitz equivalence on K . In this paper, we formulate a broad class of hyperbolic graphs, called expansive hyperbolic graphs, to capture the most essential properties from the augmented trees and the hyperbolic boundaries (e.g., the special geodesics, bounded degree property, metric doubling property, and Hölder equivalence). We also study a new setup of "weighted" IFS and investigate its connection with the self-similar energy form in the analysis of fractals.

We calculate the Gromov--Hausdorff distance between a line segment and a circle in the Euclidean plane. To do that, we introduced a few new notions like round spaces and nonlinearity degree of a metric space.

For any metric space X , finite subset spaces of X provide a sequence of isometric embeddings X=X(1)⊂X(2)⊂⋯ . The existence of Lipschitz retractions r n :X(n)→X(n−1) depends on the geometry of X in a subtle way. Such retractions are known to exist when X is an Hadamard space or a finite-dimensional normed space. But even in these cases it was unknown whether the sequence { r n } can be uniformly Lipschitz. We give a negative answer by proving that Lip( r n ) must grow with n when X is a normed space or an Hadamard space.

It is observed that a natural analog of the Hahn-Banach theorem is valid for metric functionals but fails for horofunctions. Several statements of the existence of invariant metric functionals for individual isometries and 1-Lipschitz maps are proved. Various other definitions, examples and facts are pointed out related to this topic. In particular it is shown that the metric (horofunction) boundary of every infinite Cayley graphs contains at least two points.

A closed convex polytope in n dimensions defined by m linear inequality constraints is considered. If L is a straight line drawn in any direction from any feasible point P, then in general, it intersects every constraint at one point, giving m intersections. It is shown that there exists a unique feasible point Q somewhere along this line, such that the sum of 1/di values is 0, where di is the algebraic distance between Q and the intersection with constraint i, measured along the line. The point Q is defined as the harmonic point along the line L. The harmonic center of the polytope is defined as that point which is the harmonic point for all n lines drawn through it, each parallel to one of the coordinate axes. The existence and uniqueness of such a center is shown. The harmonic center can be calculated using the coordinate search algorithm (CS), as illustrated with some simple examples. The harmonic center defined here is a generalization of the BI center defined earlier and is better in several respects. It is shown that the harmonic center of the polytope is also the harmonic point for any line drawn through it in any direction. It is also shown that for any strictly feasible point P, there exists a unique harmonic hyperplane passing through it, such that P is the harmonic point for any line which lies in the harmonic hyperplane and passes through P.