A rational linear combination of Chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the Euler and Pontryagin numbers. In dimension at least 3, only multiples of the top Chern number, which is the Euler characteristic, are invariant under diffeomorphisms that are not necessarily orientation preserving. In the ...

Abstract. The main purpose of this paper is to prove an identity of symmetry for the Frobenius-Euler polynomials. It turns out that the recurrence relation and multiplication theorem for the Frobenius-Euler polynomials which discussed in [ K. Shiratani, S. Yamamoto, On a p-adic interpolation function for the Euler numbers and its derivatives, Memo. Fac. Sci. Kyushu University Ser.A, 39(1985), 1...

The present study investigates the question of whether and how Euler diagrams can improve syllogistic reasoning. Previous research reports that there is no evidence to show the psychological advantage of Euler diagrams in solving syllogisms. In our experiments, we use Euler diagrams with “named points”, which are formulated within the simple Euler system introduced in our previous work. The dia...

This paper consists of two parts. In the first auxilliary part, we deal with sets with cyclic order. For any such set O we introduce following [BG] a cocycle l : G × G → Z valued in {0,±1} ⊂ Z on the group G of automorphisms of O. The cohomology class of l in H(G,Z) will be called the Euler class. If K is an ordered field, then the projective line P(K) has a cyclic order and PSL2(K) acts orderp...

Recently, λ-Bernoulli and λ-Euler numbers are studied in [5, 10]. The purpose of this paper is to present a systematic study of some families of the q-extensions of the λBernoulli and the λ-Euler numbers by using the bosonic p-adic q-integral and the fermionic p-adic q-integral. The investigation of these λ-q-Bernoulli and λ-q-Euler numbers leads to interesting identities related to these objec...

Area-proportional Euler diagrams have many applications, for example they are often used for visualizing data in medical and biological domains. There have been a number of recent research efforts to automatically draw Euler diagrams when the areas of the regions are not considered, leading to a range of different drawing techniques. By contrast, substantially less progress has been made on the...

Euler diagrams are often used for visualizing data collected into sets. However, there is a significant lack of guidance regarding graphical choices for Euler diagram layout. To address this deficiency, this paper asks the question ‘does the shape of a closed curve affect a user’s comprehension of an Euler diagram?’ By empirical study, we establish that curve shape does indeed impact on underst...

3D Euler diagrams visually represent the set-theoretic notions of intersection, containment and disjointness by using closed, orientable surfaces. In previous work, we introduced 3D Venn and Euler diagrams and formally defined them. In this paper, we consider the drawability of data sets using 3D Venn and Euler diagrams. The specific contributions are as follows. First, we demonstrate that ther...

In this thesis, we propose a method for the visualisation of overlapping sets and of fuzzy graph clusterings based on Euler diagrams. Euler diagrams are probably the most intuitive and most used method to depict sets in which elements can be shared. Such a powerful visualisation metaphor could be an invaluable visualisation tool, but the automatic generation of Euler diagrams still presents man...

The purpose of this paper is to illustrate the relationship between the topological property of the Euler characteristic and a combinatorial object, the Möbius function, in the context of finite T0-spaces. To do this I first explain the fundamental connection between such spaces and finite partially ordered sets by proving some facts fundamental to the study of finite spaces. Then I define the ...