Related papers: A curious identity and the volume of the root sphe…
The goal of these lectures is to explain speaker's results on uniqueness properties of spherical varieties. By a uniqueness property we mean the following. Consider some special class of spherical varieties. Define some combinatorial…
There is an elegant expression for the volume of hypercube $[0,1]^n$ clipped by a single hyperplane. In the article the formula is generalized to the case of more than one hyperplane. An important foundation for the result is Lawrence's…
We investigate the curves in the complex plane which are generated by sequences of real numbers being the lifts of the points on the orbit of an orientation preserving circle homeomorphism. Geometrical properties of these curves such as…
The congruence orbit of a matrix has a natural connection with the linear complementarity problem on simplicial cones formulated for the matrix. In terms of the two approaches -- the congruence orbit and the family of all simplicial cones…
By means of the generating function method, a linear recurrence relation is explicitly resolved. The solution is expressed in terms of the Stirling numbers of both the first and the second kind. Two remarkable pairs of combinatorial…
We give formulas for the number of polynomials over a finite field with given root multiplicities, in particular in cases when the formula is surprisingly simple (a power of q). Besides this concrete interpretation, we also prove an…
The geometry of closed surfaces equipped with a Euclidean metric with finitely many conical points of arbitrary angle is studied. The main result is that the set of closed geodesics is dense in the space of geodesics.
The intrinsic volumes are measures of the content of a convex body. This paper uses probabilistic and information-theoretic methods to study the sequence of intrinsic volumes of a convex body. The main result states that the intrinsic…
The unique real root of cos(x) = x, recently referred to as the Dottie number, is expressed as an iteral of cosine. Using the derivatives of iterals, it is shown why this number is achieved starting from any real number, when the iterates…
We give closed form expressions for the numbers of multi-rooted plane trees with specified degrees of root vertices. This results in an infinite number of integer sequences some of which are known to have an alternative interpretation. We…
Complements of union-closed families of sets, over a finite ground set, are known as simply rooted families of sets. Cubical sets are widely studied topological objects having applications in computational homology. In this paper, we look…
Brion proved that the valuation cone of a complex spherical variety is a fundamental domain for a finite reflection group, called the little Weyl group. The principal goal of this paper is to generalize this fundamental theorem to fields of…
We define a multiplication on the surreal numbers as higher inductive-inductive types.
We prove several formulas for the distribution of positive roots.
We study the cone of effective divisors and the total coordinate ring of wonderful varieties, with applications to their automorphism group. We show that the total coordinate ring of any spherical variety is obtained from that of the…
We describe a straightforward construction of the pseudo-split absolutely pseudo-simple groups of minimal type with irreducible root systems of type $BC_n$; these exist only in characteristic $2$. We also give a formula for the dimensions…
The convex hull of N independent random points chosen on the boundary of a simple polytope in R^n is investigated. Asymptotic formulas for the expected number of vertices and facets, and for the expectation of the volume difference are…
We classify the singularities of a surface ruled by conics: they are rational double points of type $A_n$ or $D_n$. This is proved by showing that they arise from a precise series of blow-ups of a suitable surface geometrically ruled by…
For divisors over smooth projective varieties, we show that the volume can be characterized by the duality between pseudo-effective cone of divisors and movable cone of curves. Inspired by this result, we give and study a natural…
We propose a version of the volume conjecture that would relate a certain limit of the colored Jones polynomials of a knot to the volume function defined by a representation of the fundamental group of the knot complement to the special…