Differentiability of solutions of some evolution equations in nonreflexive Banach spaces

Summary The purpose of this paper is to present a regularity result that provides a unified treatment of the Cauchy problem for certain nonlinear partial differential equations that appear in kinetic theory. Part. 1 contain the main theorem, based on the theory of evolution equations. In part 2 it is indicated how these abstract results are applicable to the spatially homogeneous Boltzmann equazion and to the kinetic equation of vehicular traffic. Entrata in Redazione il 28 giugno 1978. Work performed under the auspices of the G.N.A.F.A. of the National Research Couucil.     

Catalytic enantioselective conjugate addition of dialkylzinc reagents to N-substituted-2,3-dehydro-4-piperidones

The first, highly enantioselective, copper/phosphoramidite-catalyzed conjugate addition of dialkylzinc reagents to N-substituted 2,3-dehydro-4-piperidones is described.     

Integral Theory for Hopf Algebroids

The theory of integrals is used to analyze the structure of Hopf algebroids. We prove that the total algebra of a Hopf algebroid is a separable extension of the base algebra if and only if it is a semi-simple extension and if and only if the Hopf algebroid possesses a normalized integral. It is a Frobenius extension if and only if the Hopf algebroid possesses a nondegenerate integral. We give also a sufficient and necessary condition in terms of integrals, under which it is a quasi-Frobenius extension, and illustrate by an example that this condition does not hold true in general. Our results are generalizations of classical results on Hopf algebras.     

Polymerization in liquid crystals—XI: Copolymerization of N-acryloyl piperidine with cholesteryl acrylate and phenyl-p-acryloyloxy benzoate

N-acryloyl piperidine (AP) was copolymerized with mesogenic monomers [such as cholesteryl acrylate (CA) and phenyl-p-acryloyloxy benzoate (PAB)] in isotropic solution using radical initiation (A_AP = 1.1, r_CA = 0.16, and r_AP = 0.81, r_PAB = 0.16. The copolymers are anisotropic having minimum values of transition and clearing points with composition. The composition corresponding to the minimum is a boundary between two structures as indicated by wide-angle X-ray scattering. Distribution of the gyration radii determined by GPC in tetrahydrofuran is identical to that of solid polymers measured by small-angle X-ray scattering, except that the average value of the latter is about 10^?6 cm less than that of the former. It was established that AP as a non-mesogenic comonomer did not suppress the mesomorphism of the copolymer and in fact formed mesomorphic homopolymers. In the solid polymer, the macromolecule which is anisotropic in geometry was found to form the structural unit having a minimum of the melting and clearing point with composition, as for a eutectic system.     

A theorem on the endomorphism ring of reduced torsion-free abelian groups and some applications

Sunto Si espone una condiztone sufficiente affinchè un anello topologico sia l'anello degli endomorfismi, dotato della topologia finita, di un gruppo abeliano ridotto, senza torsione e di cardinalità ≤. Il risultato ottenuto estende un teorema di A. L. S. Corner ([3] Theorem1.1) ed un teorema di A. Orsatti ([9] Teorema A*).

---

---

Entrata in Redazione il 17 febbraio 1977.

---

Lavoro eseguito nell'ambito dei Gruppi di Ricerca Matematica del C.N.R.     

Very weak solutions to elliptic equations with singular convection term

We study Dirichlet problem for a nonlinear equation with a drift term. Despite the presence of the singular convection term, we establish existence and uniqueness of a solution in spaces larger than the natural one.     

The weak theory of monads

We construct a ‘weak’ version EMw(K) of Lack and Street's 2-category of monads in a 2-category K, by replacing their compatibility constraint of 1-cells with the units of monads by an additional condition on the 2-cells. A relation between monads in EMw(K) and composite pre-monads in K is discussed. If K admits Eilenberg-Moore constructions for monads, we define two symmetrical notions of ‘weak liftings’ for monads in K. If moreover idempotent 2-cells in K split, we describe both kinds of weak lifting via an appropriate pseudo-functor EMw(K)→K. Weak entwining structures and partial entwining structures are shown to realize weak liftings of a comonad for a monad in these respective senses. Weak bialgebras are characterized as algebras and coalgebras, such that the corresponding monads weakly lift for the corresponding comonads and also the comonads weakly lift for the monads.