This paper introduces a new set of orthogonal moment Functions-franklin Moments (FMs). The kernel functions of FMs is Franklin set, which is a class of complete orthogonal spline function set of degree 1. The implementation of FMs does not involve any numerical approximation and has a rather low computation complexity, since the basis set has the advantages of lower order. These properties make...

Many problems in data mining can be viewed as a special case of the problem of enumerating the closed elements of an independence system with respect to some specific closure operator. Motivated by real-world applications, e.g., in track mining, we consider a generalization of this problem to strongly accessible set systems and arbitrary closure operators. For this more general problem setting,...

We prove that the countable intersection of C1-diffeomorphic images of certain Diophantine sets has full Hausdorff dimension. For example, we show this for the set of badly approximable vectors in Rd, improving earlier results of Schmidt and Dani. To prove this, inspired by ideas of McMullen, we define a new variant of Schmidt’s (α, β)-game and show that our sets are hyperplane absolute winning...

This refuted a conjecture of B. Zilber that a strongly minimal theory must either be locally modular or interpret an infinite field (see [7]). Hrushovski’s method was extended and applied to many other questions, for example to the fusion of two strongly minimal theories ([4]) or recently to the construction of a bad field in [3]. There were also attempts to simplify Hrushovski’s original const...

We propose a method for realising the proofs of Intuitionistic Zermelo-Fraenkel set theory (IZF) by strongly normalising λ-terms. This method relies on the introduction of a Curry-style type theory extended with specific subtyping principles, which is then used as a low-level language to interpret IZF via a representation of sets as pointed graphs inspired by Aczel’s hyperset theory. As a conse...

We examine the topology of the clique complexes of the graphs of weakly and strongly separated subsets of the set [n] = {1, 2, . . . , n}, which, after deleting all cone points, we denote by ∆̂ws(n) and ∆̂ss(n), respectively. In particular, we find that ∆̂ws(n) is contractible for n ≥ 4, while ∆̂ss(n) is homotopy equivalent to a sphere of dimension n− 3. We also show that our homotopy equivalences ...

Algorithms for listing the subgraphs satisfying a given property (e.g., being a clique, a cut, a cycle, etc.) fall within the general framework of set systems. A set system (U ,F) uses a ground set U (e.g., the network nodes) and an indicator F ⊆ 2U of which subsets of U have the required property. For the problem of listing all sets in F maximal under inclusion, the ambitious goal is to cover ...