Usually we the number the points Let x l ,x , . . .,xn be n distinct points in a metric space . will restrict ourselves to the plane . Denote by D(x , . . .,x ) of distinct distances determined by x , . . .,x . Assume lthat n are in r-dimensional space . Denote byl n f (n)= min D(x, . . .,x), r xl, . . .,x l n n I conjectured more than 40 years ago that f (n)> c n/(log n) 2 . The lattice points...

We describe a framework that combines a combinatorial approach, pattern matching and automated deduction to generate and solve geometry problems for high school mathematics. Such a system would help teachers to quickly generate large numbers of questions on a geometry topic. Students can explore and revise specific topics covered in classes and textbooks based on generated questions. The system...

let $r$ be an infinite ring. here we prove that if $0_r$ belongs to ${x_1x_2cdots x_n ;|; x_1,x_2,dots,x_nin x}$ for every infinite subset $x$ of $r$, then $r$ satisfies the polynomial identity $x^n=0$. also we prove that if $0_r$ belongs to ${x_1x_2cdots x_n-x_{n+1} ;|; x_1,x_2,dots,x_n,x_{n+1}in x}$ for every infinite subset $x$ of $r$, then $x^n=x$ for all $xin r$.

Geometry representations in production Monte Carlo radiation transport codes used for linear-transport simulations are traditionally limited to combinatorial geometry (CG) topologies. While CG representations of input geometries are efficient to query, they are difficult to construct. In the Integrated-TIGER-Series (ITS) Monte Carlo code suite, a new approach for radiation transport geometry en...

We investigate the relationships between the Lipschitz outer geometry and the embedded topological type of a hypersurface germ in (Cn, 0). It is well known that the Lipschitz outer geometry of a complex plane curve germ determines and is determined by its embedded topological type. We prove that this does not remain true in higher dimensions. Namely, we give two normal hypersurface germs (X1, 0...

Hilbert geometry is a metric geometry that extends the hyperbolic Cayley-Klein geometry. In this video, we explain the shape of balls and their properties in a convex polygonal Hilbert geometry. First, we study the combinatorial properties of Hilbert balls, showing that the shapes of Hilbert polygonal balls depend both on the center location and on the complexity of the Hilbert domain but not o...

A portrait is a combinatorial model for discrete dynamical system on finite set. We study the geometry of moduli spaces, whose points correspond to equivalence classes point configurations affine line which there exist polynomials realizing dynamics given portrait. present results and pose questions inspired by large-scale computational survey intersections spaces in low degree.

We describe a prototype of new experimental GeoGebra command and tool, Discover, that analyzes geometric figures for salient patterns, properties, theorems. This tool is basic implementation automated discovery in elementary planar geometry. The paper focuses on the mathematical background implementation, as well methods to avoid combinatorial explosion when storing interesting properties figure.

I work in combinatorics, algebraic geometry, convex geometry and commutative algebra while staying informed on certain topics in category theory and ring theory. In particular, I focus on toric varieties and singularity theory. The study of toric varieties lies at the intersection of combinatorics, algebraic geometry, convex geometry and integer programming. There is a correspondence between ce...