Rigidity and moduli space in Real Algebraic Geometry

Let R be a real closed field and let X be an affine algebraic variety over R. We say that X is universally map rigid (UMR for short) if, for each irreducible affine algebraic variety Z over R, the set of nonconstant rational maps from Z to X is finite. A bijective map from an affine algebraic variety over R to X is called weak change of the algebraic structure of X if it is regular and φ⁻¹ is a Nash map, which preserves nonsingular points. We prove the following rigidity theorem: every affine algebraic variety over R is UMR up to a weak change of its algebraic structure. Let us introduce another notion. Let Y be an affine algebraic variety over R. We say that X and Y are algebraically unfriendly if all the rational maps from X to Y and from Y to X are trivial, i.e., Zariski locally constant. From the preceding theorem, we infer that, if dim (X)≥1, then there exists a set of weak changes of the algebraic structure of X such that, for each t,s ∈ R with t≠s, and are algebraically unfriendly. This result implies the following expected fact: For each (nonsingular) affine algebraic variety X over R of positive dimension, the natural Nash structure of X does not determine the algebraic structure of X. In fact, the moduli space of birationally nonisomorphic (nonsingular) affine algebraic varieties over R, which are Nash isomorphic to X, has the same cardinality of R. This result was already known under the special assumption that R is the field of real numbers and X is compact and nonsingular. The author is a member of GNSAGA of CNR, partially supported by MURST and European Research Training Network RAAG 2002–2006 (HPRN–CT–00271).     

Synthesis of gem-difluorinated 1,6-naphthyridine-5,7-diones

A synthetic route toward new 8,8-difluoro-1,6-naphthyridine-5,7-diones, which are of interest as new building blocks in pharmaceutical chemistry, is described. The key steps include a copper-mediated cross-coupling of ethyl bromodifluoroacetate and 2-bromo-3-cyanopyridine, followed by hydrolysis of the nitrile function and subsequent cyclization.     

Stability of a self-gravitating homogeneous resistive plasma

In this paper, we analyze the stability of a homogeneous self-gravitating plasma, having a non-zero resistivity. This study provides a generalization of the Jeans paradigm for determining the critical scale above which gravitational collapse is allowed.We start by discussing the stability of an ideal self-gravitating plasma embedded in a constant magnetic field. We outline the existence of an anisotropic feature of the gravitational collapse. In fact, while in the plane orthogonal to the magnetic field the Jeans length is enhanced by the contribution of the magnetic pressure, outside this plane perturbations are governed by the usual Jeans criterion. The anisotropic collapse of a density contrast is sketched in detail, suggesting that the linear evolution provides anisotropic initial conditions for the non-linear stage, where this effect could be strongly enforced.The same problem is then faced in the presence of non-zero resistivity and the conditions for the gravitational collapse are correspondingly extended. The relevant feature emerging in this resistive scenario is the cancelation of the collapse anisotropy in weakly conducting plasmas. In this case, the instability of a self-gravitating resistive plasma is characterized by the standard isotropic Jeans length in any directions. The limit of very small resistivity coefficient is finally addressed, elucidating how reminiscence of the collapse anisotropy can be found in the different values of the perturbation frequency inside and outside the plane orthogonal to the magnetic field.     

Nd:YVO₄ amplifier for ultrafast low-power lasers

An Nd:YVO₄ amplifier consisting of two modules end pumped at 808?nm at 30?W total absorbed power has been designed for efficient, diffraction-limited amplification of ultrafast pulses from low-power seeders. We investigated amplification with a 50?mW, 7?ps Nd:YVO₄ oscillator, a 2?mW, 15?ps Yb fiber laser, and a 30?mW, 300?fs Nd:glass laser. Output power as high as 9.5?W with 8?ps pulses was achieved with the 250?MHz vanadate seeder, whereas the 20?MHz fiber laser was amplified to 6?W. The femtosecond seeder allowed extracting Fourier-limited 4?ps pulses at 7?W output power. To our knowledge, these are the shortest pulses from any Nd:YVO₄ laser device with at least 7?W output power. This suggests a novel approach to exploit the gain bandwidth of vanadate amplifiers with high output power levels. Such amplifier technology promises to offer an interesting alternative to high-power thin disk oscillators at few picoseconds duration, as well as to regenerative amplifiers with low-repetition-rate fiber seeders.     

Small Strain Mechanical Properties of Latex-Based Nanocomposite Films

A waterborne latex-based technique, in which functionalized laponite is attached to PS and acrylic latex particles, is used to prepare films containing up to 50 wt% laponite. At high laponite contents this leads to a cellular arrangement of the laponite-rich layers, concentrated at the latex particle interfaces. MDSC shows that a significant proportion of the organic matrix does not contribute to the glass transition. However, this “rigid” matrix fraction arises from intercalation of the laponite stacks, and cannot account for the large increases in global stiffness in the rubbery state (T > T_g) on laponite addition. The mechanical response for T > T_g is therefore discussed in terms of a four-phase structure, in which the intercalated laponite stacks embedded in a matrix with a relatively high rubbery modulus form a cellular structure, which is in turn embedded in a matrix whose modulus is closer to that of the bulk polymer. The importance of the cellular arrangement is underlined by the much lower rubbery modulus observed by DMA in specimens produced by deforming the original films in plane strain compression to produce oriented textures with relatively little connectivity between the laponite-rich layers.     

Solving a class of low rank d.c. programs via a branch and bound approach: A computational experience

Various classes of d.c. programs have been studied in the recent literature due to their importance in applicative problems. In this paper we consider a branch and bound approach for solving a class of d.c. problems. Both stack policies and partitioning rules are analyzed, pointing out their performance effectiveness by means of the results of a computational experience.     

Some remarks on Baire’s grand theorem

We provide a game theoretical proof of the fact that if f is a function from a zero-dimensional Polish space to \( \mathbb N^{\mathbb N}\) that has a point of continuity when restricted to any non-empty compact subset, then f is of Baire class 1. We use this property of the restrictions to compact sets to give a generalisation of Baire’s grand theorem for functions of any Baire class.     

2-Imidazolines. V. The formation of hydroxy derivatives in the synthesis of 4,5-diamino-2-imidazolines from 1,2-diimmonium salts and amidines

During the preparation of 1-aryl- and 1-acyl-4,5-dipiperidino-4,5-dihydroimidazoles (1) from diimmonium salts (3), derivatives in which one of the piperidino groups was substituted by an hydroxy group were often isolated. The structure and the genesis of these compounds are discussed.     

Decarbonylation of complex iridium carbonyls in solution: A problem in chemical dynamics

The kinetics of decarbonylation of [Ir(CO)(dp)₂]Cl and [IrCl(CO)₂(Ph₃P)₂] has been studied in different solvents, at temperatures between ?25° and +70°C, by means of reactors of defined fluid dynamics which allow a separation to be made between “physical” and “chemical” rate constants. Chemical rate constants have been found to depend markedly on the diffusion coefficients of carbon monoxide in the various solvents. The process of decarbonylation has been described, for both reactions, by the sequence: structural isomerization, characterized by a very low preexponential factor, decomposition of the less stable isomer against the solvent's barrier, and diffusion of carbon monoxide to the gas–liquid interface. The kinetic problems involved in the determination of rate constants and their implications have been emphasized.