The Shapley value for cooperative games under precedence constraints

Cooperative games are considered where only those coalitions of players are feasible that respect a given precedence structure on the set of players. Strengthening the classical symmetry axiom, we obtain three axioms that give rise to a unique Shapley value in this model. The Shapley value is seen to reflect the expected marginal contribution of a player to a feasible random coalition, which allows us to evaluate the Shapley value nondeterministically. We show that every exact algorithm for the Shapley value requires an exponential number of operations already in the classical case and that even restriction to simple games is #P-hard in general. Furthermore, we outline how the multi-choice cooperative games of Hsiao and Raghavan can be treated in our context, which leads to a Shapley value that does not depend on pre-assigned weights. Finally, the relationship between the Shapley value and the permission value of Gilles, Owen and van den Brink is discussed. Both refer to formally similar models of cooperative games but reflect complementary interpretations of the precedence constraints and thus give rise to fundamentally different solution concepts.     

Medicinal Organometallic Chemistry: Designing Metal Arene Complexes as Anticancer Agents

The field of medicinal inorganic chemistry is rapidly advancing. In particular organometallic complexes have much potential as therapeutic and diagnostic agents. The carbon‐bound and other ligands allow the thermodynamic and kinetic reactivity of the metal ion to be controlled and also provide a scaffold for functionalization. The establishment of structure–activity relationships and elucidation of the speciation of complexes under conditions relevant to drug testing and formulation are crucial for the further development of promising medicinal applications of organometallic complexes. Specific examples involving the design of ruthenium and osmium arene complexes as anticancer agents are discussed.     

Genetically Encoded Bioluminescent Indicator for ERK2 Dimer in Living Cells

In this study, a genetically encoded bioluminescent indicator for ERK2 dimer was developed with the split Renilla luciferase complementation method, in which the formation of ERK2 dimer induces a spontaneous emission of bioluminescence in living cells. In response to extracellular stimuli, such as epidermal growth factor (EGF) or 17β‐estradiol (E2), extracellular signal‐regulated kinase 2 (ERK2) is phosphorylated by its upstream kinase MEK, and also phosphorylates its substrates in various regions of the cell, including the nucleus. Phosphorylated ERK2 is led to form its dimer, thereby transporting itself into the nucleus. We demonstrated with the indicator that stimulation with EGF or E2 induces the formation of ERK2 dimer in living MCF‐7 cells. The dynamics of this dimer formation was examined and discussed.     

Weak Attractive Ligand–Polymer and Related Interactions in Catalysis and Reactivity: Impact,Applications, and Modeling

The notion of weak attractive ligand–polymer interactions is introduced, and its potential application, importance, and conceptual links with “cooperative” ligand–substrate interactions are discussed. Synthetic models of weak attractive ligand–polymer interactions are described, in which intramolecular weak C? H???F? C interactions (the existence of which remains contentious) have been detected by NMR spectroscopy and neutron and X‐ray diffraction experiments. These C? H???F? C interactions carry important implications for the design of catalysts for olefin polymerization, because they provide support for the practical feasibility of ortho‐F???H_β ligand–polymer contacts proposed for living Group 4 fluorinated phenoxyimine catalysts. The notion of weak attractive noncovalent interactions between an “active” ligand and the growing polymer chain is a novel concept in polyolefin catalysis.     

Untersuchungen über Papaverin

Ohne Zusammenfassung     

Lattice path combinatorics and asymptotics of multiplicities of weights in tensor powers

We give asymptotic formulas for the multiplicities of weights and irreducible summands in high-tensor powers V_λ^⊗N of an irreducible representation V_λ of a compact connected Lie group G. The weights are allowed to depend on N, and we obtain several regimes of pointwise asymptotics, ranging from a central limit region to a large deviations region. We use a complex steepest descent method that applies to general asymptotic counting problems for lattice paths with steps in a convex polytope.     

Prädikatives und funktionales Denken in der Wahrscheinlichkeitsrechnung

In order to emphasize functional thinking in mathematical education, arguments are brought forward for an early and frequent use of tree diagrams in teaching stochastics. Since set and tree diagrams can be regarded as two sides of the same matter, Bayes’ theorem, which is strongly associated with predicative set diagrams, need not be dealt with explicitly any longer. Set diagrams are replaced by tree diagrams that are labelled in detail and by tree inversion as a functional instrument of dealing with conditioned probabilities. This technique is demonstrated by three typical problems in the context of conditioned probability. Finally, two letters by Pascal to Fermat are analysed in order to illustrate that predicative and functional approaches were already pursued in the early stages of probability calculus in the 17^th century.