By Alberto Cialdea, Flavia Lanzara, Paolo Emilio Ricci

ISBN-10: 3764398973

ISBN-13: 9783764398972

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This booklet is dedicated to Hilbert's twenty first challenge (the Riemann-Hilbert challenge) which belongs to the speculation of linear structures of normal differential equations within the complicated area. the matter concems the lifestyles of a Fuchsian approach with prescribed singularities and monodromy. Hilbert used to be confident that the sort of approach continually exists.

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2), f, g are given continuous functions, q > 1 , cq = 1q 1 − 1q and λ ≥ 0 represents a discount rate. Two cases of interest for the applications are when τx = inf t ≥ 0 : Xt ∈ Rn \ Ω or (the exit time problem) τx = +∞ (the state-constrained problem). The optimal control problem associated to these data is to minimize, for each given initial position X0 = x, the functional J(x, a) with respect to feedback controls a belonging to a speciﬁed set Ax of admissible controls. 1) with q p = q−1 . 4) at any point x0 ∈ Ω and for all C function Ψ touching from below the graph of u at x0 (the supersolution condition).

The purpose of this paper is to formulate an alternative mathematical model for the electron and positron, a model which is geometrically much simpler. The advantage of our approach is that it does not require the use of spinors, Pauli matrices or covariant diﬀerentiation. The only geometric concepts we use are those of a • • • • metric, diﬀerential form, wedge product, exterior derivative. Our model overcomes the logical problem of distinguishing the electron from the positron: these correspond to clockwise and anticlockwise rotations of the coframe.

4, 583–630. L. Lions, R´esolution de probl`emes elliptiques quasilin´eaires, (French) Arch. Rational Mech. Anal. 74 (1980), no. 4, 335–353. L. -London (1982). L. Lions, Quelques remarques sur les probl`emes elliptiques quasilin´eaires du second ordre, (French) J. Analyse Math. 45 (1985), 234–254. L. E. Souganidis, Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 22 (2005), no. 5, 667–677. [17] O.