On the density of the space of continuous and uniformly continuous functions

For X a metrizable space and (Y,ρ) a metric space, with Y pathwise connected, we compute the density of (C(X,(Y,ρ)),σ)—the space of all continuous functions from X to (Y,ρ), endowed with the supremum metric σ. Also, for (X,d) a metric space and (Y,‖⋅‖) a normed space, we compute the density of (UC((X,d),(Y,ρ)),σ) (the space of all uniformly continuous functions from (X,d) to (Y,ρ), where ρ is the metric induced on Y by ‖⋅‖). We also prove that the latter result extends only partially to the case where (Y,ρ) is an arbitrary pathwise connected metric space.To carry such an investigation out, the notions of generalized compact and generalized totally bounded metric space, introduced by the author and A. Barbati in a former paper, turn out to play a crucial rôle. Moreover, we show that the first-mentioned concept provides a precise characterization of those metrizable spaces which attain their extent.     

Graphical operations on projective spaces

Motivated by the work of Crapo and Rota [6] on the lifting of a projective complex, we introduce a class of invariant operations associated to integral-weighted graphs, which we call graphical operations. Such operations generalize the sixth harmonic of a quadranguler set on a projective line. We determine the expansion of the graphical operations in terms of multi-linear bracket polynomials in a Grassmann-Cayley algebra. Reducibility and compositions of such invariant operations are also investigated with a number of examples.Supported by Courant Instructorship, New York University.     

Low temperature synthesis, structure and bioactivity of xCaO· (1−x)SiO2 glasses

Glasses of composition expressed by the following general formula xCaO·(1?x)SiO₂ (0.30≤x≤0.50) can be prepared by means of the sol-gel route starting from tetramethylorthosilicate and calcium nitrate tetrahydrate They are all difficult to prepare by means of the traditional technique of quenching the melt, because of the high liquidus temperature,T ₁ t that in the case ofx=0.3 glass isT ₁ =1650°C. The DTA apparatus appears a valuable tool for defining the procedure necessary to obtain the glass through the sol-gel route. The glassx=0.3 is bioactive. The experimental results suggest that the gel structures, such as obtained at room temperature, are very similar; only at high temperature do the reactions of hydrolysis and polycondensation go to completion and the structural units characteristic of each glass are obtained.     

An alternative for moduli stabilisation

The one-loop vacuum energy is explicitly computed for a class of perturbative string vacua where supersymmetry is spontaneously broken by a T-duality invariant asymmetric Scherk–Schwarz deformation. The low-lying spectrum is tachyon-free for any value of the compactification radii and thus no Hagedorn-like phase-transition takes place. Indeed, the induced effective potential is free of divergence, and has a global anti-de Sitter minimum where geometric moduli are naturally stabilised.     

Thermal evolution of ferromagnetic metallic glasses

A traditional TG apparatus was modified by placing two permanent magnets producing a controlled magnetic field (TG(M): Magneto Thermogravimetry). This technique proved to be useful to study both structural relaxation and crystallisation of ferromagnetic metallic glasses. Results obtained for the amorphous alloys Fe₄₀Ni₄₀P₁₄B₆ and Fe_62.5Co₆Ni_7.5Zr₆Nb₂Cu₁B₁₅, are reported in this paper. Structural relaxation can be evaluated by measuring changes in Curie temperature induced by thermal treatments. Crystallisation in TG(M) is detected through a change in the measured apparent mass (difference between the sample mass and magnetic force driving it upward). These results were confirmed by DSC analysis. Whether the obtained crystalline phase is ferromagnetic, it can be identified through its Curie temperature, measured by TG(M). In fact the value of 770°C measured as Curie temperature of crystallised Fe_62.5Co₆Ni_7.5Zr₆Nb₂Cu₁B₁₅led to conclude that the only ferromagnetic crystalline phase is a-Fe. These hypothesis was confirmed by XRD analysis, showing that the first crystallisation yields to a-Fe nanocrystals. This revised version was published online in July 2006 with corrections to the Cover Date.     

Thermal reactions of tributyltin hydride with alpha-azido esters: unexpected intervention of tin triazene adducts under both nonradical and radical conditions

Thermal reaction of various alpha-azido esters with Bu(3)SnH in refluxing benzene results in smooth production of 3-(tributylstannyl)-1-triazene adducts affording cyclized 1,2,3-triazol-4-ones in preference to reduced amines and thence provides a new useful method for the preparation of these triazole derivatives. In the presence of AIBN the occurrence of triazene products still remains important or even exclusive and, consequently, generation of the expected stannylaminyl radicals is seriously limited. With 2-azidomalonates and alpha-azido-beta-keto esters stannyltriazenes can similarly occur in the absence of the radical initiator, but in the latter cases the ensuing triazenes undergo preferential cyclization onto the ketone moiety to give reactive hydroxytriazolines. Contrary to alpha-azido esters, in the presence of AIBN alpha-azido-beta-keto esters as well as azidomalonates give rise only to the usual stannylaminyl radicals. A possible explanation for the different behavior of the mono- and dicarbonyl azides in the presence of AIBN is put forward.     

The inelastic hard dimer gas: a nonspherical model for granular matter

We study a two-dimensional gas of inelastic smooth hard dimers. Since the collisions between dimers are dissipative, being characterized by a coefficient of restitution alpha<1, and no external driving force is present, the energy of the system decreases in time and no stationary state is achieved. However, the resulting nonequilibrium state of the system displays several interesting properties in close analogy with systems of inelastic hard spheres, whose relaxational dynamics has been thoroughly explored. We generalize to inelastic systems a recently method introduced [G. Ciccotti and G. Kalibaeva, J. Stat. Phys. 115, 701 (2004)] to study the dynamics of rigid elastic bodies made up of different spheres held together by rigid bonds. Each dimer consists of two hard disks of diameter d, whose centers are separated by a fixed distance a. By describing the rigid bonds by means of holonomic constraints and deriving the appropriate collision rules between dimers, we reduce the dynamics to a set of equations which can be solved by means of event-driven simulation. After deriving the algorithm we study the decay of the total kinetic energy, and of the ratio between the rotational and the translational kinetic energy of inelastic dimers. We show numerically that the celebrated Haff's homogeneous cooling law t(-2), describing how the kinetic energy of an inelastic hard-sphere system with a constant coefficient of restitution decreases in time, holds even in the case of these nonspherical particles. We fully characterize this homogeneous decay process in terms of appropriate decay constants and confirm numerically the scaling behavior of the velocity distributions.     

Intramolecular cyclization of acyl radicals onto the azido group: a new radical approach to cyclized lactams

[reaction: see text] Aryl- and alkyl-derived azidoacyl radicals, generated from thiolesters by intramolecular homolytic substitution at the sulfur, can undergo five- and six-membered cyclization onto the azido moiety to give cyclized lactams.