An example of a superproportional graph

In the theory of random graphs, several classes of graphs occur that are exceptional cases in limit theorems for subgraph counts. The purpose of this article is to show the existence of graphs in one of these classes, by providing an explicit, computer generated, example. We also show that the class is closed under complementation. © 1994 John Wiley & Sons, Inc.     

Aqua­(phthalocyaninato)magnesium n‐propyl­amine disolvate

The crystals of the title compound, [Mg(C₃₂H₁₆N₈)(H₂O)]·2C₃H₉N, are built up from MgPc(H₂O) [Pc is phthalo­cyaninate(2−)] and n‐propyl­amine mol­ecules that inter­act via O—H⋯N hydrogen bonds. The MgPc(H₂O) mol­ecule is non‐planar. The central Mg atom is coordinated by the four equatorial isoindole N atoms of the Pc ring system and by the O atom of an axial water mol­ecule. The Mg atom is displaced by 0.509 (1) Å from the N₄ plane towards the water O atom. MgPc(H₂O)·2(n‐propyl­amine) mol­ecules related by the inversion centre are linked by N—H⋯O hydrogen bonds to form a dimeric aggregate.     

Convex ordering criteria for Lévy processes

Modelling financial and insurance time series with Lévy processes or with exponential Lévy processes is a relevant actual practice and an active area of research. It allows qualitatively and quantitatively good adaptation to the empirical statistical properties of asset returns. Due to model incompleteness it is a problem of considerable interest to determine the dependence of option prices in these models on the choice of pricing measures and to establish nontrivial price bounds. In this paper we review and extend ordering results of stochastic and convex type for this class of models. We also extend the ordering results to processes with independent increments (PII) and present several examples and applications as to α-stable processes, NIG-processes, GH-distributions, and others. Criteria are given for the Lévy measures which imply corresponding comparison results for European type options in (exponential) Lévy models.     

Solving the Gleason problem on linearly convex domains

Let be a bounded, connected linearly convex set in with boundary. We show that the maximal ideal (both in ) and ) consisting of all functions vanishing at is generated by the coordinate functions . Received: 2 July 2001; in final form: 26 September 2001 / Published online: 28 February 2002     

A multivalued version of Sharkovskiĭ’s theorem holds with at most two exceptions

A multivalued version of Sharkovskiĭ’s theorem is formulated for M-maps on linear continua and, more generally, for triangular M-maps on a Cartesian product of linear continua. This improves the main result of [AP1] in the sense that our multivalued analogue holds with at most two exceptions. A further specification requires some additional restrictions. For instance, 3- orbits of m-maps imply the existence of k-orbits for all k ? \mathbbNk \in {\mathbb{N}} , except possibly for k ?k \in {4, 6}. It is also shown that, on every connected linearly ordered topological space, an M-map with orbits of all periods can always be constructed. This demonstrates that Baldwin’s classification of linear continua in terms of admissible periods [Ba] is useless for multivalued maps.