Uniqueness of post-gelation solutions of a class of coagulation equations

We prove well-posedness of global solutions for a class of coagulation equations which exhibit the gelation phase transition. To this end, we solve an associated partial differential equation involving the generating functions before and after the phase transition. Applications include the classical Smoluchowski and Flory equations with multiplicative coagulation rate and the recently introduced symmetric model with limited aggregations. For the latter, we compute the limiting concentrations and we relate them to random graph models.     

An analytic approach to existence and uniqueness for martingale problems in infinite dimensions

We prove existence and uniqueness for a class of martingale problems in a Hilbert space. We solve the associated Kolmogorov equation and prove that the corresponding semigroup is determined by a kernel of measures if a Schauder-type regularity is satisfied. Received: 18 May 1998 / Revised version: 27 September 1999 / Published online: 5 September 2000     

Convergence of approximations of monotone gradient systems

We consider stochastic differential equations in a Hilbert space, perturbed by the gradient of a convex potential. We investigate the problem of convergence of a sequence of such processes. We propose applications of this method to reflecting O.U. processes in infinite dimension, to stochastic partial differential equations with reflection of Cahn-Hilliard type and to interface models. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

I fondamenti chimici e chimico-fisici della microdeterminazione colorimetrica dell'acido fosforico mediante l'azzurro di molibdeno

Résumé Après une énumération brève des méthodes principales pour la détermination de l'acide phosphorique dans une matière biologique, on a conclu, que la méthode colorimétrique pour la détermination du bleu de molybdène est la méthode la plus digne de confiance pour la détermination de l'acide phosphorique.On discute les bases théoriques de cette méthode et les conditions chimiques et physico-chimiques dont l'accomplissement est nécessaire pour atteindre des dosages précises. Aussi on met en ligne de compte l'application de cette méthode sur le dosage indirect du magnésium. Enfin on communique le mode d'exécution pratique pour la détermination de l'acide phosphorique et du magnésium dans une matière biologique, prenant pour base les conditions optimales, trouvées au cours du travail.

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Summary The principal methods for the determination of phosphoric acid in biological material are briefly enumerated, and the conclusion is reached that the colorimetric determination as molybdenum blue is the most reliable.The theoretical basis underlying the method is discussed, and the chemical and physico-chemical conditions enumerated which are necessary for the attainment of accurate results; the extension of the method to the indirect determination of magnesium is also considered. In conclusion, a practical procedure is proposed, based on the most favourable theoretical conditions, for the determination of both phosphoric acid and magnesium in biological material.

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Zusammenfassung Nach einer kurzen Aufzählung der hauptsächlichsten Methoden zur Bestimmung der Phosphorsäure in biologischem Material wird der Schluß gezogen, daß unter allen Methoden die kolorimetrische zur Bestimmung als Molybdänblau die vertrauenswürdigste ist.Es werden die theoretischen Grundlagen behandelt, auf denen diese Methode beruht, und die chemischen und physikochemischen Bedingungen diskutiert, deren Erfüllung zur Erreichung genauer Bestimmungen notwendig ist. Es wird auch die Ausdehnung der Methode auf die indirekte Bestimmung des Magnesiums in Betracht gezogen. Schließlich wird unter Zugrundelegung der als optimal festgestellten Bedingungen die praktische Ausführungsform sowohl für die Bestimmung der Phosphorsäure wie des Magnesiums in biologischem Material vorgeschlagen.

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Rissunto Fatta una rapida ressegna dei principali metodi di dosaggio dell'acido fosforico in materiale biologico, si conclude che fra tutti il metodo colorimetrico all'azzurro di molibdeno é quello che da maggior affidamento.Si trattano i fondamenti teorici su cui si basa tale metodo e si discutono le condizioni chemiiche e chimico-fisiche necessarie par arrivare a determinazioni esatte. Si contempla pure l'estensione del metodo al dosaggio indiretto del magnesio. Infine tenendo conto delle condizioni optimum stabilite, si propone il procedimento pratico di determinazione sia dell'acido fosforico sia del magnesio in materiale biologico.

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Diese Arbeit wurde noch von der früheren Schriftleitung der Mikrochemie redigiert.     

Quasi-invariance properties of a class of subordinators

We study absolute-continuity relationships for a class of stochastic processes, including the gamma and the Dirichlet processes. We prove that the laws of a general class of non-linear transformations of such processes are locally equivalent to the law of the original process and we compute explicitly the associated Radon–Nikodym densities. This work unifies and generalizes to random non-linear transformations several previous quasi-invariance results for gamma and Dirichlet processes.     

Approximate maximizers of intricacy functionals

G. Edelman, O. Sporns, and G. Tononi introduced in theoretical biology the neural complexity of a family of random variables. A previous work showed that this functional is a special case of intricacy, i.e., an average of the mutual information of subsystems with specific mathematical properties. Moreover, its maximum value grows at a definite speed with the size of the system. In this work, we compute exactly this speed of growth by building “approximate maximizers” subject to an entropy condition. These approximate maximizers work simultaneously for all intricacies. We also establish some properties of arbitrary approximate maximizers, in particular the existence of a threshold in the size of subsystems of approximate maximizers: most smaller subsystems are almost equidistributed, most larger subsystems determine the full system. The main ideas are a random construction of almost maximizers with a high statistical symmetry and the consideration of entropy profiles, i.e., the average entropies of sub-systems of a given size. The latter gives rise to interesting questions of probability and information theory.     

Integration by parts on Bessel Bridges and related stochastic partial differential equationsIntegration par parties sur Ponts de Bessel et EDPS correspondantes

We prove integration by parts formulae with respect to the law of Bessel Bridges of dimension δ?3. For δ=3 we have an infinite-dimensional boundary measure, and for δ>3 a singular logarithmic derivative. We give applications to SPDEs with additive space-time white noise and singular drifts, whose solutions are non-negative. To cite this article: L. Zambotti, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 209–212.