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.     

Possible application of X-ray optical elements for reducing the spectral bandwidth of an X-ray SASE FEL

A new design for a single pass X-ray Self-Amplified Spontaneous Emission (SASE) FEL is proposed. The scheme consists of two undulators and an X-ray monochromator located between them. The first stage of the FEL amplifier operates in the SASE linear regime. After the exit of the first undulator the electron bunch is guided through a non-isochronous bypass and the X-ray beam enters the monochromator. The main function of the bypass is to suppress the modulation of the electron beam induced in the first undulator. This is possible because of the finite value of the natural energy spread in the beam. At the entrance to the second undulator the radiation power from the monochromator dominates significantly over the shot noise and the residual electron bunching. As a result the second stage of the FEL amplifier operates in the steady-state regime when the input signal bandwidth is small with respect to that of the FEL amplifier. Integral losses of the radiation power in the monochromator are relatively small because grazing incidence optics can be used. The proposed scheme is illustrated for the example of the 6 nm option SASE FEL at the TESLA Test Facility under construction at DESY. As shown in this paper the spectral bandwidth of such a two-stage SASE FEL (Δλ/λ 5 × 10⁻⁵) is close to the limit defined by the finite duration of the radiation pulse. The average brilliance is equal to 7 × 10²⁴ photons/(s × mrad² × mm² × 0.1% bandw.) which is by two orders of magnitude higher than the value which could be reached by the conventional SASE FEL. The monochromatization of the radiation is performed at a low level of radiation power (about 500 times less than the saturation level) which allows one to use conventional X-ray optical elements (grazing incidence grating and mirrors) for the monochromator design.     

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.