Independence of inadmissible strategies and best reply stability: a direct proof

Hillas (1990) introduced a definition of strategic stability based on perturbations of the best reply correspondence that satisfies all of the requirements given by Kohlberg and Mertens (1986). Hillas et al. (2001) point out though that the proofs of the iterated dominance and forward induction properties were not correct. They also provide a proof of the IIS property, a stronger version of both iterated dominance and forward induction, using the results of that paper. In this note we provide a direct proof of the IIS property.Received February 2002     

Stable rank of corner rings

B. Blackadar recently proved that any full corner in a unital C*-algebra has K-theoretic stable rank greater than or equal to the stable rank of . (Here is a projection in , and fullness means that .) This result is extended to arbitrary (unital) rings in the present paper: If is a full idempotent in , then . The proofs rely partly on algebraic analogs of Blackadar's methods and partly on a new technique for reducing problems of higher stable rank to a concept of stable rank one for skew (rectangular) corners . The main result yields estimates relating stable ranks of Morita equivalent rings. In particular, if where is a finitely generated projective generator, and can be generated by elements, then .

     

Self-similar fragmentations derived from the stable tree II: splitting at nodes

We study a natural fragmentation process of the so-called stable tree introduced by Duquesne and Le Gall, which consists in removing the nodes of the tree according to a certain procedure that makes the fragmentation self-similar with positive index. Explicit formulas for the semigroup are given, and we provide asymptotic results. We also give an alternative construction of this fragmentation, using paths of Lévy processes, hence echoing the two alternative constructions of the standard additive coalescent by fragmenting the Brownian continuum random tree or using Brownian paths, respectively due to Aldous-Pitman and Bertoin.Mathematics Subject Classification (2000): 60J25, 60G52Acknowledgement Many thanks to Jean Bertoin for many precious comments on this work, and to Jean-François Le Gall for discussions related to the stable tree. Thanks also to an anonymous referee for a careful reading and very helpful comments that helped to consequently improve the presentation of this work.     

Comodules and Landweber exact homology theories

We show that, if E is a commutative MU-algebra spectrum such that is Landweber exact over , then the category of -comodules is equivalent to a localization of the category of -comodules. This localization depends only on the heights of E at the integer primes p. It follows, for example, that the category of -comodules is equivalent to the category of -comodules. These equivalences give simple proofs and generalizations of the Miller-Ravenel and Morava change of rings theorems. We also deduce structural results about the category of -comodules. We prove that every -comodule has a primitive, we give a classification of invariant prime ideals in , and we give a version of the Landweber filtration theorem.     

On a functional equation generalizing the class of semistable distributions

With ?(p),p≥0 the Laplace-Stieltjes transform of some infinitely divisible probability distribution, we consider the solutions to the functional equation ?(p-e ^?pβΠ _i=1 ^m ?^γi (c _i p) for somem≥1,c _i>0, γ _i >0,i=1., …,m, β ε ®. We supply its complete solutions in terms of semistable distributions (the ones obtained whenm=1). We then show how to obtain these solutions as limit laws (r → ∞) of normalized Poisson sums of iid samples when the Poisson intensity λ(r) grows geometrically withr.     

Authenticity control of essential oils containing citronellal and citral by chiral and stable-isotope gas-chromatographic analysis

Enantioselective capillary GC on a Supelco β-DEX 225 column (heptakis(2,3-di-O-acetyl-6-O-tert-butyldimethylsilyl)-β-cyclodextrin SPB 20poly—20% diphenyl, 80% dimethylsiloxane) and isotope-ratio mass spectrometry, coupled online with capillary GC on an HP5 column have been used for origin-specific analysis and authenticity control of essential oils, for example lemon (Citrus limon), lemongrass (Cymbopogon citratus and Cymbopogon flexuosus), citronella (Cymbopogon nardus L.—Ceylon type and Cymbopogon winterianus—Java type), Litsea cubeba, Lippia citriodora, lemon myrtle (Backhousia citriodora), lemon gum (Eucalyptus citriodora), and, especially, precious lemon balm oil (Melissa officinalis L.). Isotope data (δ¹³C_PDB and δ²H_V-SMOW) for citral (neral + geranial) and citronellal from on-line GC–C/Py–IRMS and chiral data for citronellal in these essential oils are reported. The possibility of using these data to determine the origin of these essential oils and to detect adulteration is discussed. Principal-components analysis (PCA) of specific compounds in two essential oils of lemongrass and Litsea cubeba was performed as a practical statistical method for distinguishing between these two types of oil.     

Establishing a chromium-reactor design for measuring δ2H values of solid polyhalogenated compounds using direct elemental analysis and stable isotope ratio mass spectrometry

²H/¹H isotope ratios of polyhalogenated compounds were determined by elemental analysis and isotope ratio mass spectrometry (EA-IRMS). Initial measurements with standard EA-IRMS equipment, which used high-temperature pyrolysis to convert the organic compounds into hydrogen, did not achieve significant signals for polychlorinated pesticides and related compounds, presumably due to the formation of HCl instead of hydrogen. To reverse this problematic reaction, a chromium reactor was incorporated into the element analyzer system, which scavenged Cl, forming chromium chloride and releasing hydrogen again in the form of H₂. The optimized system therefore allowed the δ²H values of polyhalogenated compounds to be determined. A quality assurance program was developed based on several parameters. (i) Each compound was analyzed using a sequence of five injections, where the first measurement was discarded. (ii) Recovery of H (when calculated relative to acetanilide) had to be >90% for all replicates in a sequence. (iii) All δ-values within a sequence had to vary by less than 10‰. (iv) Results had to be reproducible on another day with a different sample scheme. Once this reproducibility had been established, variabilities in the δ²H values of organohalogen standards were investigated using the technique. The highest δ²H value of +75‰ was found for o,p′-DDD, whereas the strongest depletion in deuterium was found for Melipax (–181‰). The most important results for comparable compounds were as follows. DDT-related compounds gave δ²H values of between +59 and +75‰ (technical DDT, o,p′- and p,p′-DDD) or in the range of approximately −1‰, indicative of the different sources/methods of producing this compound. Four HCH isomers from the same supplier showed relatively similar hydrogen isotope distributions, whereas two lindane (γ–HCH) standards from other sources had 39‰ less deuterium. This difference is likely due to different purification steps during the isolation of pure lindane from the technical HCH mixture. An even greater difference was observed between the δ²H values of Toxaphene (US product dating from 1978) and Melipax (product from the former East Germany, dating from 1979), which gave δ²H values of –101‰ and –181‰, respectively, meaning that both products were easily distinguished via δ²H-IRMS. Fractioning of hydrogen isotopes in the atmospheric water cycle was suggested as one reason for the different values. In this theory, the water (which had different δ²H values depending on where it was taken from) was incorporated during the biosynthesis of camphene, which is the natural product used to produce both products. These results indicate that hydrogen isotope-specific analysis can be a valuable tool for tracing the origins of a compound in certain cases.     

C and N stable isotope variation in urine and milk of cattle depending on the diet

The stable carbon and nitrogen isotopic composition of urine and milk samples from cattle under different feeding regimes were analysed over a period of six months. The isotope ratios were measured with isotope ratio mass spectrometry (IRMS). The δ ¹³C values of milk and urine were dependent on different feeding regimes based on C₃ or C₄ plants. The δ ¹³C values are more negative under grass feeding than under maize feeding. The δ ¹³C values of milk are more negative compared to urine and independent of the feeding regime. Under grass feeding the analysed milk and urine samples are enriched in ¹³C relative to the feed, whereas under maize feeding the ¹³C/¹²C ratio of urine is in the same range and milk is depleted in ¹³C relative to the diet. The difference between the ¹⁵N/¹⁴N ratios for the two feeding regimes is less pronounced than the ¹³C/¹²C ratios. The δ ¹⁵N values in urine require more time to reach the new equilibrium, whereas the milk samples show no significant differences between the two feeding regimes.     

On the class of flat stable theories

A new notion of independence relation is given and associated to it, the class of flat theories, a subclass of strong stable theories including the superstable ones is introduced. More precisely, after introducing this independence relation, flat theories are defined as an appropriate version of superstability. It is shown that in a flat theory every type has finite weight and therefore flat theories are strong. Furthermore, it is shown that under reasonable conditions any type is non-orthogonal to a regular one. Concerning groups in flat theories, it is shown that type-definable groups behave like superstable ones, since they satisfy the same chain condition on definable subgroups and also admit a normal series of definable subgroup with semi-regular quotients.     

A Nontrivial Homotopy Element of Order p2 Detected by the Classical Adams Spectral Sequence

Let p be an odd prime. The authors detect a nontrivial element ã _p of order p² in the stable homotopy groups of spheres by the classical Adams spectral sequence. It is represented by \(a_0^{p - 2} h_1 \in Ext_A^{p - 1,pq + p - 2} (\mathbb{Z}/p,\mathbb{Z}/p)\) in the E₂-term of the ASS and meanwhile p · ã ^p is the first periodic element α ^p .