Sul postulato della completezza lineare dell'Hilbert

Sunto Si disente criticamente il detto postulato, con particolare riguardo al problema della sua compatibilità entro il sistema ipotetico — deduttivo dell'Hilbert. A Giovanni Sansone nel suo 70^mo compleanno.     

Sulle successioni di funzioni quasi continue negli spazi astratti

Sunto Si studia una definizione del tutto generale di funzione quasi continua; si enuncia e dimostra poi un teorema, concettualmente nuovo, per le successioni di funzioni che sono quasi continue nel senso generale studiato. A Mauro Picone nel suo 70^mo compleanno. Lavoro eseguito presso l'Istituto Nazionale per le Applicazioni del Calcolo.     

Su alcuni problemi fortemente ben posti di controllo ottimo

Summary Let P_n be a sequence of optimal control problems with fixed end times (described by ordinary differential equations, with pointwise and norm (of controls) constraints, and with general functional). Let P₀ be an ? unperturbed ? problem (of the same type). In this paper theorems are obtained about the strong convergence (in some L^p) of optimal controls of P_n to some optimal control of P₀, about the uniform convergence x_n → x₀ of states, and the property thatmin P_n min P₀. As corollaries, convergence theorems for some calculus of variations problems can be derived. Weak convergence theorems of optimal controls of P_n to an optimal one of P₀ were considered in[7]. A general abstract theorem about strong convergence of minimum points, generalizing a result in[5], is proved.

---

---

Indirizzo dell'autore: Istituto di Matematica, via L. B. Alberti 4 - 16132 Genova. Lavoro eseguito nell'ambito del Centro di Matematica e di Fisica Teorica del C.N.R. presso l'Università di Genova.

---

Entrata in Redazione il 4 maggio 1972.     

Role of surface integrals in the Hamiltonian formulation of general relativity

It is shown that if the phase space of general relativity is defined so as to contain the trajectories representing solutions of the equations of motion then, for asymptotically flat spaces, the Hamiltonian does not vanish but its value is given rather by a nonzero surface integral. If the deformations of the surface on which the state is defined are restricted so that the surface moves asymptotically parallel to itself in the time direction, then the surface integral gives directly the energy of the system, prior to fixing the coordinates or solving the constraints. Under more general conditions (when asymptotic Poincaré transformations are allowed) the surface integrals giving the total momentum and angular momentum also contribute to the Hamiltonian. These quantities are also identified without reference to a particular fixation of the coordinates. When coordinate conditions are imposed the associated reduced Hamiltonian is unambiguously obtained by introducing the solutions of the constraints into the surface integral giving the numerical value of the unreduced Hamiltonian. In the present treatment there are therefore no divergences that cease to be divergences after coordinate conditions are imposed. The procedure of reduction of the Hamiltonian is explicity carried out for two cases: (a) Maximal slicing, (b) ADM coordinate conditions.A Hamiltonian formalism which is manifestly covariant under Poincaré transformations at infinity is presented. In such a formalism the ten independent variables describing the asymptotic location of the surface are introduced, together with corresponding conjugate momenta, as new canonical variables in the same footing with the g_ij, π^ij. In this context one may fix the coordinates in the “interior” but still leave open the possibility of making asymptotic Poincaré transformations. In that case all ten generators of the Poincaré group are obtained by inserting the solution of the constraints into corresponding surface integrals.     

Influence of the substituent on selective photocatalytic oxidation of aromatic compounds in aqueous TiO2 suspensions

Experimental results are reported showing that the photocatalytic oxidation of aromatic compounds containing an electron-donor group (EDG) gives rise mainly to ortho- and para-monohydroxy derivatives while in the presence of an electron-withdrawing group (EWG) all the monohydroxy derivatives are obtained.     

Trees, wreath products and finite Gelfand pairs

We present a new construction of finite Gelfand pairs by looking at the action of the full automorphism group of a finite spherically homogeneous rooted tree of type r on the variety V(r,s) of all spherically homogeneous subtrees of type s.This generalizes well-known examples as the finite ultrametric space, the Hamming scheme and the Johnson scheme.We also present further generalizations of these classical examples. The first two are based on Harary's notions of composition and exponentiation of group actions. Finally, the generalized Johnson scheme provides the inductive step for the harmonic analysis of our main construction.