61,036 research outputs found

    Determination of stability constants using genetic algorithms

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    A genetic algorithm (GA)-simplex hybrid approach has been developed for the determination of stability constants using calorimetric and polarographic data obtained from literature sources. The GA determined both the most suitable equilibrium model for the systems studied and the values of the stability constants and the heats of formation for the calorimetric studies. As such, a variable length chromosome format was devised to represent the equilibrium models and stability constants (and heats of formation). The polarographic data were obtained from studies of cadmium chloride and lead with the crown ether dicyclohexyl-18-crown-6. The calorimetric data were obtained from a study of a two step addition reaction of Hg(CN)2 with thiourea. The stability constants obtained using the GA-simplex hybrid approach compare favourably with the values quoted in the literature

    Local minimality of the volume-product at the simplex

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    It is proved that the simplex is a strict local minimum for the volume product, P(K)=min(vol(K) vol(K^z)), K^z is the polar body of K with respect to z, the minimum is taken over z in the interior of K, in the Banach-Mazur space of n-dimensional (classes of ) convex bodies. Linear local stability in the neighborhood of the simplex is proved as well. The proof consists of an extension to the non-symmetric setting of methods that were recently introduced by Nazarov, Petrov, Ryabogin and Zvavitch, as well as proving results of independent interest, concerning stability of square order of volumes of polars of non-symmetric convex bodies.Comment: Mathematika, accepte

    Standard Simplices and Pluralities are Not the Most Noise Stable

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    The Standard Simplex Conjecture and the Plurality is Stablest Conjecture are two conjectures stating that certain partitions are optimal with respect to Gaussian and discrete noise stability respectively. These two conjectures are natural generalizations of the Gaussian noise stability result by Borell (1985) and the Majority is Stablest Theorem (2004). Here we show that the standard simplex is not the most stable partition in Gaussian space and that Plurality is not the most stable low influence partition in discrete space for every number of parts k3k \geq 3, for every value ρ0\rho \neq 0 of the noise and for every prescribed measures for the different parts as long as they are not all equal to 1/k1/k. Our results do not contradict the original statements of the Plurality is Stablest and Standard Simplex Conjectures in their original statements concerning partitions to sets of equal measure. However, they indicate that if these conjectures are true, their veracity and their proofs will crucially rely on assuming that the sets are of equal measures, in stark contrast to Borell's result, the Majority is Stablest Theorem and many other results in isoperimetric theory. Given our results it is natural to ask for (conjectured) partitions achieving the optimum noise stability.Comment: 14 page

    Strengthened inequalities for the mean width and the ℓ-norm

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    Barthe proved that the regular simplex maximizes the mean width of convex bodies whose John ellipsoid (maximal volume ellipsoid contained in the body) is the Euclidean unit ball; or equivalently, the regular simplex maximizes the ℓ ‐norm of convex bodies whose Löwner ellipsoid (minimal volume ellipsoid containing the body) is the Euclidean unit ball. Schmuckenschläger verified the reverse statement; namely, the regular simplex minimizes the mean width of convex bodies whose Löwner ellipsoid is the Euclidean unit ball. Here we prove stronger stability versions of these results. We also consider related stability results for the mean width and the ℓ ‐norm of the convex hull of the support of centered isotropic measures on the unit sphere

    Continuity and Equilibrium Stability

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    This paper discusses the problem of stability of equilibrium points in normal form games in the tremling-hand framework. An equilibrium point is called perffect if it is stable against at least one seqence of trembles approaching zero. A strictly perfect equilibrium point is stable against every such sequence. We give a sufficient condition for a Nash equilibrium point to be strictly perfect in terms of the primitive characteristics of the game (payoffs and strategies), which is new and not known in the literature. In particular, we show that continuity of the best response correspondence (which can be stated in terms of the primitives of the game) implies strict perfectness; we prove a number of other useful theorems regarding the structure of best responce correspondence in normal form games.Strictly perfect equilibrium, best responce correspondence, unit simplex, face of a unit simplex

    Homological stability for families of Coxeter groups

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    We prove that certain families of Coxeter groups and inclusions W1W2...W_1\hookrightarrow W_2\hookrightarrow... satisfy homological stability, meaning that in each degree the homology H(BWn)H_\ast(BW_n) is eventually independent of nn. This gives a uniform treatment of homological stability for the families of Coxeter groups of type AnA_n, BnB_n and DnD_n, recovering existing results in the first two cases, and giving a new result in the third. The key step in our proof is to show that a certain simplicial complex with WnW_n-action is highly connected. To do this we show that the barycentric subdivision is an instance of the 'basic construction', and then use Davis's description of the basic construction as an increasing union of chambers to deduce the required connectivity.Comment: 16 page
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