Analytic continuation

In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where the infinite series representation which initially defined the function becomes divergent. The step-wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies (defining more than one value). They may alternatively have to do with the presence of singularities. The case of several complex variables is rather different, since singularities then need not be isolated points, and its investigation was a major reason for the development of sheaf cohomology. Initial discussion Suppose f is an analytic function defined on a non-empty open subset U of the complex plane \Complex. If V is a large

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Numerical analysis of optimization-constrained differential equations

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The modeling of a system composed by a gas phase and organic aerosol particles, and its numerical resolution are studied. The gas-aerosol system is modeled by ordinary differential equations coupled with a mixed-constrained optimization problem. This coupling induces discontinuities when inequality constraints are activated or deactivated. Two approaches for the solution of the optimization-constrained differential equations are presented. The first approach is a time splitting scheme together with a fixed-point method that alternates between the differential and optimization parts. The ordinary differential equations are approximated by the Crank-Nicolson scheme and a primal-dual interior-point method combined with a warm-start strategy is used to solve the minimization problem. The second approach considers the set of equations as a system of differential algebraic equations after replacing the minimization problem by its first order optimality conditions. An implicit 5th-order Runge-Kutta method (RADAU5) is then used. Both approaches are completed by numerical techniques for the detection and computation of the events (activation and deactivation of inequality constraints) when the system evolves in time. The computation of the events is based on continuation techniques and geometric arguments. Moreover the first approach completes the computation with extrapolation polynomials and sensitivity analysis, whereas the second approach uses dense output formulas. Numerical results for gas-aerosol system made of several chemical species are proposed for both approaches. These examples show the efficiency and accuracy of each method. They also indicate that the second approach is more efficient than the first one. Furthermore theoretical examples show that the method for the computation of the activation is of second order for the first approach and exact for the second one.

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On the non-analytic behaviour of thermodynamic potentials at first order phase transitions

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Ordinary differential equations are coupled with mixed constrained optimization problems when modeling the thermodynamic equilibrium of a system evolving with time. A particular application arises in the modeling of atmospheric particles. Discontinuity points are created by the activation/deactivation of inequality constraints. A numerical method for the solution of optimization-constrained differential equations is proposed by coupling an implicit Runge-Kutta method (RADAU5), with numerical techniques for the detection of the events (activation and deactivation of constraints). The computation of the events is based on dense output formulas, continuation techniques, and geometric arguments. Numerical results are presented for the simulation of the time-dependent equilibrium of organic atmospheric aerosol particles, and show the efficiency and accuracy of the approach.

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