An orientation-preserving recurrent homeomorphism of the twosphere which is not the identity is shown to admit exactly two fixed points. A recurrent homeomorphism of a compact surface with negative Euler characteristic is periodic.

In the case of isothetic simple polyhedra there are only six different types of 3D angles. This article states and proofs a formula about counts of these angles. This complements formulas in combinatorial topology such as Euler's polyhedron formula, or the previously known formula on angle counts for isothetic polygons. The latter formula and the shown equality for angle counts of isothetic sim...

It is well-known that the maximal size of minimally 3-connected graphs of order 7 ≥ n is 9 3 − n . In this paper, we shall prove that if G is a minimally 3-connected graph of order n, and is embedded in a closed surface with Euler characteristic , χ then G contains at most } 2 , 2 min{ 2 χ − n edges. This bound is best possible for every closed surface.

In this paper we study the tangent spaces of the smooth nested Hilbert scheme Hilb n,n−1 (A 2) of points in the plane, and give a general formula for computing the Euler characteristic of a T 2-equivariant locally free sheaf on Hilb n,n−1 (A 2). Applying our result to a particular sheaf, we conjecture that the result is a polynomial in the variables q and t with non-negative integer coefficient...

Let A be a special biserial algebra over an algebraically closed field. We show that the first Hohchshild cohomology group of A with coefficients in the bimodule A vanishes if and only if A is representation finite and simply connected (in the sense of Bongartz and Gabriel), if and only if the Euler characteristic of Q equals the number of indecomposable non uniserial projective injective A-mod...

Abstract. Using the spectral dynamics, we study the critical threshold phenomena in the multidimensional restricted Euler (RE) equations. We identify sub-critical and sup-critical initial data for all space dimensions, which extends the previous result for the 3D and 4D restricted Euler equations. Our result suggests that: if the number of dimensions is odd, the finite time blowup is generic; i...

This paper considers the Euler tour of an uncertain random graph, in which some edges exist with some degrees in uncertain measure and others exist with some degrees in probability measure. The concept of Euler index of an uncertain random graph is proposed first, and then a method to calculate the Euler index of an uncertain random graph is given. In addition, some properties of the Euler inde...

Let S be a compact, oriented surface with negative Euler characteristic and let f : S → S be a homeomorphism isotopic to the identity. If there exists a periodic orbit with a non-zero rotation vector (~ p, q), then there exists a simple braid with the same rotation vector.

The absolute order is a natural partial order on a Coxeter group W . It can be viewed as an analogue of the weak order on W in which the role of the generating set of simple reflections in W is played by the set of all reflections in W . By use of a notion of constructibility for partially ordered sets, it is proved that the absolute order on the symmetric group is homotopy Cohen–Macaulay. This...

Euler discovered the pentagonal number theorem in 1740 but was not able to prove it until 1750. He sent the proof to Goldbach and published it in a paper that finally appeared in 1760. Moreover, Euler formulated another proof of the pentagonal number in his notebooks theorem around 1750. Euler did not publish this proof or communicate it to his correspondents, probably because of the difficulty...