Measure of segments which intersect a convex body from rotational formulae

Classical problems in integral geometry and geometric probability involve the kinematic measure of congruent segments of fixed length within a convex body in R³. We give this measure from rotational formulae; that is, from isotropic plane sections through a fixed point. From this result we also obtain a new rotational formula for the volume of a convex body; which is proved to be equivalent to the wedge formula for the volume.     

The isometric extension problem between unit spheres of two separable Banach spaces

In this article, we use some analytic and geometric characters of the smooth points in a sphere to study the isometric extension problem in the separable or reflexive real Banach spaces. We obtain that under some condition the answer to this problem is affirmative.     

Universal point sets for planar three-trees

For every $n\in \mathbb{N}$, we present a set $S_n$ of $O(n^{3/2}\log n)$ points in the plane such that every planar 3-tree with n vertices has a straight-line embedding in the plane in which the vertices are mapped to a subset of $S_n$. This is the first subquadratic upper bound on the cardinality of universal point sets for planar 3-trees, as well as for the class of 2-trees and serial parallel graphs.     

Formation of lithium,sodium and potassium complexes with 1,3,5‐triamino‐1,3,5‐trideoxy‐cis‐inositol (taci)

Single crystals of (1,3‐diamino‐5‐azaniumyl‐1,3,5‐trideoxy‐cis‐inositol‐κ³O²,O⁴,O⁶)(1,3,5‐triamino‐1,3,5‐trideoxy‐cis‐inositol‐κ³O²,O⁴,O⁶)lithium(I) diiodide dihydrate, [Li(C₆H₁₆N₃O₃)(C₆H₁₅N₃O₃)]I₂·2H₂O or [Li(Htaci)(taci)]I₂·2H₂O (taci is 1,3,5‐triamino‐1,3,5‐trideoxy‐cis‐inositol), (I), bis(1,3,5‐triamino‐1,3,5‐trideoxy‐cis‐inositol‐κ³O²,O⁴,O⁶)sodium(I) iodide, [Na(C₆H₁₅N₃O₃)₂]I or [Na(taci)₂]I, (II), and bis(1,3,5‐triamino‐1,3,5‐trideoxy‐cis‐inositol‐κ³O²,O⁴,O⁶)potassium(I) iodide, [K(C₆H₁₅N₃O₃)₂]I or [K(taci)₂]I, (III), were grown by diffusion of MeOH into aqueous solutions of the complexes. The structures of the Na and K complexes are isotypic. In all three complexes, the taci ligands adopt a chair conformation with axial hydroxy groups, and the metal cations exhibit exclusive O‐atom coordination. The six O atoms of the resulting MO₆ unit define a centrosymmetric trigonal antiprism with approximate D_3d symmetry. The interligand O...O distances increase significantly in the order Li < Na < K. The structure of (I) exhibits a complex three‐dimensional network of R—NH₂—H...NH₂—R, R—O—H...NH₂—R and R—O—H...O(H)—H...NH₂—R hydrogen bonds. The structures of the Na and K complexes consist of a stack of layers, in which each taci ligand is bonded to three neighbours via pairwise O—H...NH₂ interactions between vicinal HO—CH—CH—NH₂ groups.     

Possibilistic nested logic programs and strong equivalence

In this paper, the class of possibilistic nested logic programs is introduced. These possibilistic logic programs allow us to use nested expressions in the bodies and heads of their rules. By considering a possibilistic nested logic program as a possibilistic theory, a construction of a possibilistic logic programing semantics based on answer sets for nested logic programs and the proof theory of possibilistic logic is defined. In order to define a general method for computing the possibilistic answer sets of a possibilistic nested program, the idea of equivalence between possibilistic nested programs is explored. By considering properties of equivalence between possibilistic programs, a process of transforming a possibilistic nested logic program into a possibilistic disjunctive logic program is defined. Given that our approach is an extension of answer set programming, we also explore the concept of strong equivalence between possibilistic nested logic programs. To this end, we introduce the concept of poss SE-models. Therefore, we show that two possibilistic nested logic programs are strong equivalents whenever they have the same poss SE-models.The expressiveness of the possibilistic nested logic programs is illustrated by a scenario from the medical domain. In particular, we exemplify how possibilistic nested logic programs are expressive enough for capturing medical guidelines which are pervaded by vagueness and qualitative information.