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We prove a conjecture of Meszaros and Morales on the volume of a flow polytope. Independently from our work, Zeilberger sketched a proof of their conjecture. In fact, our proof is the same as Zeilberger's proof. The purpose of this note is…

An interval in a combinatorial structure S is a set I of points which relate to every point from S I in the same way. A structure is simple if it has no proper intervals. Every combinatorial structure can be expressed as an inflation of a…

This article is studying the roots of the reliability polynomials of linear consecutive-\textit{k}-out-of-\textit{n}:\textit{F} systems. We are able to prove that these roots are unbounded in the complex plane, for any fixed $k\ge2$. In the…

We consider the relation between simplicial volume and two of its variants: the stable integral simplicial volume and the integral foliated simplicial volume. The definition of the latter depends on a choice of a measure preserving action…

The celebrated Poncelet porism is usually studied for a pair of smooth conics that are in a general position. Here we discuss Poncelet porism in the real plane - affine or projective, when that is not the case, i.e. the conics have at least…

In this paper, we use multivariate splines to investigate the volume of polytopes. We first present an explicit formula for the multivariate truncated power, which can be considered as a dual version of the famous Brion's formula for the…

We show that the integral foliated simplicial volume of a connected compact oriented smooth manifold with a regular foliation by circles vanishes.

The Generalized Lax Conjecture asks whether every hyperbolicity cone is a section of a semidefinite cone of sufficiently high dimension. We prove that the space of hyperbolicity cones of hyperbolic polynomials of degree $d$ in $n$ variables…

In his paper "On the Schlafli differential equality", J. Milnor conjectured that the volume of n-dimensional hyperbolic and spherical simplices, as a function of the dihedral angles, extends continuously to the closure of the space of…

The average density of zeros for monic generalized polynomials, $P_n(z)=\phi(z)+\sum_{k=1}^nc_kf_k(z)$, with real holomorphic $\phi ,f_k$ and real Gaussian coefficients is expressed in terms of correlation functions of the values of the…

We give a new proof of the following conjecture of Yoccoz: the sum of the logarithm of the conformal radius of fixed Siegel disks of monic quadratic polynomials and of the Brjuno function of their rotation number is bounded from above. In a…

We propose a study of the foliations of the projective plane induced by simple derivations of the polynomial ring in two indeterminates over the complex field. These correspond to foliations which have no invariant algebraic curve nor…

Let $V_{g,m,n}(\overrightarrow L,\overrightarrow \theta)$ be the Weil-Petersson volume of the moduli space of hyperbolic surfaces of genus g with m geodesic boundary components of length $\overrightarrow L=(\ell_1,...,\ell_m)$ and $n$ cone…

The intrinsic volumes of a convex cone are geometric functionals that return basic structural information about the cone. Recent research has demonstrated that conic intrinsic volumes are valuable for understanding the behavior of random…

For any polynomial f with complex coefficients we find a remarkable subset of poles of the motivic zeta function. It is combinatorially determined by any log resolution and it admits an intrinsic interpretation in terms of contact loci of…

For n greater than or equal to 4, the square of the volume of an n-simplex satisfies a polynomial relation with coefficients depending on the squares of the areas of 2-faces of this simplex. First, we compute the minimal degree of such…

"[M]athematicians care no more for logic than logicians for mathematics." Augustus de Morgan, 1868. Proofs are traditionally syntactic, inductively generated objects. This paper presents an abstract mathematical formulation of propositional…

We show that for any closed Riemannian manifold with dimension at least two and with nonpositive curvature, if it admits an isolated, closed totally geodesic submanifold of codimension one, then its simplicial volume is positive. As a…

We give a combinatorial proof of an identity that involves Eulerian numbers and was obtained algebraically by Brenti and Welker (2009). To do so, we study alcoved triangulations of dilated hypersimplices. As a byproduct, we describe the…

We define a combinatorial object that can be associated with any conic-line arrangement with ordinary singularities, which we call the combinatorial Poincar\'e polynomial. We prove a Terao-type factorization statement on the splitting of…