Related papers: On the Blaschke conjecture for 3-webs

Based on the theory of invariant sets of descending flow, we give a new proof of the existence of three nontrivial solutions and some remarks on it.

In a recent paper, Bilu et al. studied a conjecture of Marques and Lengyel on the $p$-adic valuation of the Tribonacci sequence. In this article, we study the $p$-adic valuation of third order linear recurrence sequences by considering a…

We give a simple axiomatic definition of a rational-valued invariant s(W,V,e) of triples (W,V,e), where W is a (smooth, oriented, closed) 6-manifold and V is a 3-submanifold of W, and where e is a second rational cohomology class of the…

Monsky proved that a square cannot be dissected into an odd number of triangles of equal area. Stein conjectured that the same holds for any polygon whose edges can be paired into parallel and equal-length segments. We prove Stein's…

We consider 3-webs, hyper-para-complex structures and integrable Segre structures on manifolds of even dimension and generalise the second heavenly Pleba\'nski equation in the context of higher-dimensional hyper-para-complex structures. We…

We prove two mixed versions of the Discrete Nodal Theorem of Davies et. al. [3] for bounded degree graphs, and for three-connected graphs of fixed genus $g$. Using this we can show that for a three-connected graph satisfying a certain…

We give a necessary condition for two diagrams of $3$-regular spatial graphs with the same underlying abstract graph $G$ to represent isotopic spatial graphs. The test works by reading off the writhes of the knot diagrams coming from a…

We compute all third-order local invariants accessible via randomised measurements and employ them to derive separability criteria. The reconstruction of the invariants yields experimentally accessible entanglement criteria for multipartite…

Thomassen conjectured that triangle-free planar graphs have an exponential number of $3$-colorings. We show this conjecture to be equivalent to the following statement: there exists a positive real $\alpha$ such that whenever $G$ is a…

We continue the development of methods for enumerating nodal curves on smooth complex surfaces, stressing the range of validity. We illustrate the new methods in three important examples. First, for up to eight nodes, we confirm…

We discuss conjectures on Hamiltonicity in cubic graphs (Tait, Barnette, Tutte), on the dichromatic number of planar oriented graphs (Neumann-Lara), and on even graphs in digraphs whose contraction is strongly connected (Hochst\"attler). We…

A connected graph G is 3-flow-critical if G does not have a nowhere-zero 3-flow, but every proper contraction of G does. We prove that every n-vertex 3-flow-critical graph other than K_2 and K_4 has at least 5n/3 edges. This bound is tight…

The skewness of a graph G is the minimum number of edges in G whose removal results in a planar graph. By appropriately introducing a weight to each edge of a graph, we determine, among other thing, the skewness of the generalized Petersen…

In this paper we prove a conjecture about the dimension of linear systems of surfaces of degree d in P^3 through at most eight multiple points in general position.

In an Oberbeck-Boussinesq model, rigorously derived, which includes compressibility, one could expect that the onset of convection for the B\'enard problem occurs at a higher critical Rayleigh number. Since of the difficulties related to…

We study the maximal number of triangulations that a planar set of $n$ points can have, and show that it is at most $30^n$. This new bound is achieved by a careful optimization of the charging scheme of Sharir and Welzl (2006), which has…

An explicit polynomial in the linking numbers $l_{ij}$ and Milnor's triple linking numbers $\mu(rst)$ on six component links is shown to be a well-defined finite type link-homotopy invariant. This solves a problem raised by B. Mellor and D.…

The purpose of this paper is to combine classical methods from transcendental number theory with the technique of restriction to real scalars. We develop a conceptual approach relating transcendence properties of algebraic groups to results…

The question of Lorentz invariance for finite N approximations of relativistic membranes is addressed. We find that one of the classical manifestations of Lorentz-invariance is not possible for NxN matrices (at least when N=2 or 3). How the…

We give a probabilistic construction of a $3$-uniform hypergraph on $N$ vertices with independence number $O(\log N / \log \log N)$ in which there are at most two edges among any four vertices. This bound is tight and solves a longstanding…