In this work, we present a family of exact self-similar solutions for the distribution of enclosed areas under mean curvature flow on the union of disjoint closed curves in the plane. To our knowledge, this is the first example of closed-form exact self-similar solutions for the mean curvature flow. We perform numerical simulations for large sets of initial data obtained as the zero level set o...

We call a family G ⊂ P[n] a k-generator of P[n] if every x ⊂ [n] can be expressed as a union of at most k disjoint sets in G. Frein, Lévêque and Seb˝ o [1] conjectured that for any n ≥ k, such a family must be at least as large as the k-generator obtained by taking a partition of [n] into classes of sizes as equal as possible, and taking the union of the power-sets of the classes. We generalize...

We prove that any family of compact convex sets in R which satisfy the (p, q)-property (p ≥ q ≥ d+1) can be pierced with Õ((pq) q−1 q−d ) points for d ≥ 3 and O((pq) q−1 q−2 ) for d = 2. This improves (already for d = 2 and q = 3) the previously best known bound (of Õ(p 2+d)) provided in Alon and Kleitman’s celebrated proof of the Hadwiger Debrunner conjecture (for the case q = d+ 1). We also p...

Background: To compare the union times of the uncomplicated tibial shaft fractures, which were distally locked by twocoronal and one sagittal screws and by only two coronal screws.Methods: 45 patients with tibial shaft treated with intramedullary nailing included in this study. 23 of 45 fractureswere treated with uniplanar two distal interlocking (Group 1) and 22 fractures were treated with bip...

In this paper we deal with the problem of extending Zadeh's operators on fuzzy sets (FSs) to interval-valued (IVFSs), set-valued (SVFSs) and type-2 (T2FSs) fuzzy sets. Namely, it is known that seeing FSs as SVFSs, or T2FSs, whose membership degrees are singletons is not order-preserving. We then describe a family of lattice embeddings from FSs to SVFSs. Alternatively, if the former singleton vi...

We follow 8] in asking when a set of ordinals X is a countable union of sets in K, the core model. We show that, analogously to L, an X closed under the canonical 1 Skolem function for K can be so decomposed, provided K is such that no !-closed lters are put on its measure sequence, but not otherwise. This proviso holds if there is no inner model of a weak Erd} os type property.

We study concepts of decidability (recursivity) for subsets of Euclidean spaces R k within the framework of approximate computability (type two theory of ef-fectivity). A new notion of approximate decidability is proposed and discussed in some detail. It is an eeective variant of F. Hausdorr's concept of resolvable sets and modiies and generalizes notions of recursivity known from computable an...

We specify a definable decomposition of the upper semilattice of recursively enumerable (r.e.) degrees R as the disjoint union of an ideal M and a strong filter NC. The ideal M consists of 0 together with all degrees which are parts of r.e. minimal pairs, and thus the degrees in NC are called noncappable degrees. Furthermore, NC coincides with five other apparently unrelated subclasses of R: EN...