Related papers: The Euler characteristic of the regular spherical …

A formal n-square is the set of positions in an square matrix of size n. A shuffle of a formal n-square consists of independent rotations of each row and of each column. A key result turns out to be valid at least for n <= 34 and n = 37:…

For an integer $n\geq 7$, we investigate the matroid realization space of a specific deformation of the regular $n$-gon along with its lines of symmetry. It turns out that this particular realization space is birational to the elliptic…

Let $X$ be a projective smooth surface over $\mathbb{C}$ with $H^2(\mathcal{O}_X)=0$. Let $M=M(L,\chi)$ be the moduli space of 1-dimensional semistable sheaves with determinant $\mathcal{O}_X(L)$ and Euler characteristic $\chi$. We have the…

We show that if a sequence $M_n$ of closed aspherical $d$-dimensional Riemannian manifolds with Ricci curvature uniformly bounded below and diameter uniformly bounded above collapses, then for all large enough $n$, the fundamental groups…

We express the mean Euler characteristic of a contact structure in terms of the mean indices of closed Reeb orbits for a broad class of contact manifolds, the so-called asymptotically finite contact manifolds. We show that this class is…

Consider a 3$-$dimensional manifold $N$ obtained by gluing a finite number of ideal hyperbolic tetrahedra via isometries along their faces. By varying the isometry type of each tetrahedron but keeping fixed the gluing pattern we define a…

We prove a conjecture of Cuttler et al.~[2011] [A. Cuttler, C. Greene, and M. Skandera; \emph{Inequalities for symmetric means}. European J. Combinatorics, 32(2011), 745--761] on the monotonicity of \emph{normalized Schur functions} under…

A set ${X}_{N}=\{x_1,\ldots,x_N\}$ of $N$ points on the unit sphere $\mathbb{S}^d,\,d\geq 2$ is a spherical $t$-design if the average of any polynomial of degree at most $t$ over the sphere is equal to the average value of the polynomial…

The systole of a closed Riemannian manifold is the minimal length of a non-contractible closed loop. We give a uniform lower bound for the systole for large classes of simple arithmetic locally symmetric orbifolds. We establish new bounds…

We prove that if u is a bounded smooth function in the kernel of a nonnegative Schrodinger operator $-L=-(\Delta +q)$ on a parabolic Riemannian manifold M, then u is either identically zero or it has no zeros on M, and the linear space of…

In this paper, we systematically study non-crossing chords of simple polygons in the plane. We first introduce the reduced Euler characteristic of a family of line-segments, and subsequently investigate the structure of the diagonals and…

The goal of this article is to study the space of smooth Riemannian structures on compact manifolds with boundary that satisfies a critical point equation associated with a boundary value problem. We provide an integral formula which…

The Thurston spine consists of the subset of Teichm\"uller space at which the set of shortest curves, the systoles, cuts the surface into polygons. The systole function is a topological Morse function on Teichm\"uller space. This paper…

A special generic map is a smooth map regarded as a natural generalization of Morse functions with just 2 singular points on homotopy spheres. Canonical projections of unit spheres are simplest examples of such maps and manifolds admitting…

In a closed manifold of positive dimension $n$, we estimate the expected volume and Euler characteristic for random submanifolds of codimension $r\in \{1,...,n\}$ in two different settings. On one hand, we consider a closed Riemannian…

We explicitly compute the Green's function of the spinor Klein-Gordon equation on the Riemannian and Lorentzian manifolds of the form $M_0 \times ... \times M_N$, with each factor being a space of constant sectional curvature. Our approach…

A unified construction of canonical $H^m$-nonconforming finite elements is developed for $n$-dimensional simplices for any $m, n \geq 1$. Consistency with the Morley-Wang-Xu elements [Math. Comp. 82 (2013), pp. 25-43] is maintained when $m…

A Banach space is {\it polynomially Schur} if sequential convergence against analytic polynomials implies norm convergence. Carne, Cole and Gamelin show that a space has this property and the Dunford-Pettis property if and only if it is…

Determining if a symmetric function is Schur-positive is a prevalent and, in general, a notoriously difficult problem. In this paper we study the Schur-positivity of a family of symmetric functions. Given a partition \lambda, we denote by…

We shall show that any complex minimal surface of general type with c_1^2 = 2\chi -1 having non-trivial 2-torsion divisors, where c_1^2 and \chi are the first Chern number of a surface and the Euler characteristic of the structure sheaf…