Mathematics

The Serpinsky-Knopp curve is characterized as the only curve (up to isometry) that maps a unit segment onto a triangle of a unit area, so for any pair of points in the segment, the square of the distance between their images does not exceed four times the distance between them.

We develop a new definition of fractals which can be considered as an abstraction of the fractals determined through self-similarity. The definition is formulated through imposing conditions which are governed the relation between the subsets of a metric space to build a porous self-similar structure. Examples are provided to confirm that the definition is satisfied by large class of self-similar fractals. The new concepts create new frontiers for fractals and chaos investigations.

The conjecture about the Orlicz Pólya-Szegö principle posed in [43] is proved. The cases of equality are characterized in the affine Orlicz Pólya-Szegö principle with respect to Steiner symmetrization and Schwarz spherical symmetrization.

A Heron triangle is a triangle whose side lengths and area are integers. Two Heron triangles are amicable if the perimeter of one is the area of the other. We show, using elementary techniques, that there is only one pair of amicable Heron triangles.

The Brocard porism is a known 1d family of triangles inscribed in a circle and circumscribed about an ellipse. Remarkably, the Brocard angle is invariant and the Brocard points are stationary at the foci of the ellipse. In this paper we show that a certain derived triangle spawns off a second, smaller, Brocard porism so that repeating this calculation produces an infinite, converging sequence of porisms. We also show that this sequence is embedded in a continuous family of porisms.

Steinitz's theorem states that a graph G is the edge-graph of a 3 -dimensional convex polyhedron if and only if, G is simple, plane and 3 -connected. We prove an analogue of this theorem for ball polyhedra, that is, for intersections of finitely many unit balls in R 3 .

The notion of the ultrametrics can be considered as a zero-dimensional analogue of ordinary metrics, and it is expected to prove ultrametric versions of theorems on metric spaces. In this paper, we provide ultrametric versions of the Arens--Eells isometric embedding theorem of metric spaces, the Hausdorff extension theorem of metrics, the Niemytzki--Tychonoff characterization theorem of the compactness, and the author's interpolation theorem of metrics and theorems on dense subsets of spaces of metrics.

Consider a finite collection of affine hyperplanes in R d . The hyperplanes dissect R d into finitely many polyhedral chambers. For a point x∈ R d and a chamber P the metric projection of x onto P is the unique point y∈P minimizing the Euclidean distance to x . The metric projection is contained in the relative interior of a uniquely defined face of P whose dimension is denoted by dim(x,P) . We prove that for every given k∈{0,…,d} , the number of chambers P for which dim(x,P)=k does not depend on the choice of x , with an exception of some Lebesgue null set. Moreover, this number is equal to the absolute value of the k -th coefficient of the characteristic polynomial of the hyperplane arrangement. In a special case of reflection arrangements, this proves a conjecture of Drton and Klivans [A geometric interpretation of the characteristic polynomial of reflection arrangements, Proc. Amer. Math. Soc., 138(8): 2873-2887, 2010].

We provide an interpolation theorem of a family of metrics defined on closed subsets of metrizable spaces. As an application, we observe that various sets of all metrics with properties appeared in metric geometry are dense intersections of countable open subsets in spaces of metrics on metrizable spaces. For instance, our study is applicable to the set of all non-doubling metrics and the set of all non-uniformly disconnected metrics.

The configuration space of a mechanical linkage, consisting of rigid bodies moving in space constrained by joints, is defined by algebraic conditions. If these equations do not define a complete intersection, then the dimension of the configuration space is higher than expected. These linkages violate the Chebychev-Grübler-Kutzbach formula for the degree of freedom of mobility. Mathematicians developed a variety of methods to describe and understand this phenomenon. This paper explains some of them.