On the propagation of plane waves in piezoelectromagnetic monoclinic crystals

In a piezoelectromagnetic crystalline medium belonging to the class 2 of the monoclinic crystallographic system we find some classes of piezoelectricity-induced electromagnetic waves. These are time harmonic plane waves propagating along the symmetry axis and depending only on the axial coordinate. There are two indepen- dent modes of propagation, one longitudinal and one transverse, with mechanical and electromagnetical couplings. The transverse mode admits as a particular case an electromagnetic wave with no associated elastic deformation

On discontinuity waves and smooth waves in thermo-piezoelectric bodies

The solid body B under consideration is composed of a linear thermo-piezoelectric medium, i.e., a non-magnetizable linearly elastic dielectric medium that is heat conducting and not electric conducting; B has a natural conguration, say a placement of the three-dimensional Euclidean space that B can occupy with zero stress, uniform temperature and uniform electric field. Such natural conguration and state will be used as reference. We consider processes of B constituted by small displacements, thermal deviations and small electric fields (u; T; E) superposed to the reference state. A smooth singular surface (or discontinuity surface) of order r in the triple of fields (u; T; E) is referred to as a weak (thermo-piezoelectric) wave if r >=2. Any singular surface of order r>= 2 is characteristic (for the linear thermo-piezoelectric partial dierential equations). Then smooth waves are considered. (i) It is shown that the wavefront of a plane progressive wave is characteristic if and only if the wave is isothermal. (ii) The differential equations are characterized for standing waves of a general type and for the standing waves which are sinusoidal. The latter are isothermal, isentropic, have wavefronts which are characteristic, and their directions of propagation satisfy certain constitutive conditions. (iii) The differential equations for plane progressive waves which are reversible in time are characterized

LIQUIDITY CONSTRAINTS AND SMALL FIRMS' GROWTH

There is general agreement on the difference, in terms of efficiency, between small firms in southern and northern Italy (see Prosperetti and Varetto, 1991; Giannola and Sarno, 1996; Giannola, Papagni and Sarno, 1998; Sarno, 1999). As far as technical efficiency is concerned (that is the ability to generate output from a given amount of factors of production) there appears, over the years, a persistent, huge differential (around 30 per cent) in favour of Northern firms. These findings (Giannola and Sarno, 1996; Giannola, Sarno and Papagni, 1998) show that Southern firms are not in a position to exploit economies of scale as northern firms do. The fact that firms’ size in the Mezzogiorno, for different size classes, is systematically smaller than in the rest of Italy, is a particular feature that seems to have major consequences. When the analysis shifts from technical to economic efficiency (i.e., the ability to combine the factors of production so as to equalize the weighted marginal productivity of factors), the gap between southern and northern firms in the level of production costs is dramatically reduced (ranging from 6 to 10 per cent). These results suggest that the North-South gap is not so much related to a specific inefficiency, but to the smaller size of southern firms that does not allow them to profit from returns to scale to the same extent. A possible conclusion (with relevant policy implications) is that the root of many problems of southern firms lies in the obstacles that prevent them from growing to reach an adequate operating size. A related implication is that the southern industrial structure, due to its size characteristics, is more easily affected by monetary policy and the economic cycle (Gertler and Gilchrist, 1991; 1993) The difficulties faced by southern firms become evident if we look at the evolution of the Italian manufacturing industry during the 1990s. For this purpose, we can proceed by considering a sample of firms which is representative of the entire population from a sectoral, dimensional and geographical standpoint. During the 1990s there was an overall decline in average firm size in Italian across all size classes. This is particularly marked for southern firms, in which firms’ size in 1990 was already significantly below average; by 1994 this character was considerably accentuated and was confirmed in 1997; by contrast, average northern Italian firm size showed a major increase in 1997. Moreover, the less pronounced and less persistent decline of the average size did not prevent Northern firms from expanding sales in the period in hand; this is due to the rapid growth of exports that followed the 1992 devaluation of the Italian lira. In the same period, in the Mezzogiorno, sales of local firms show a persistent and significant decline. As a consequence, the dynamics of the productivity of labour was negative (or stagnant) in the South, while it steadily grew in the rest of the country. Southern firms, while experiencing at the end of the period some significant progress in penetrating foreign markets, faced increased competition on the domestic market due to the contraction in aggregate demand, severely affected by the restrictive stance of the macroeconomic policy. All in all, these data illustrate the weakness (if not the worsening) of the competitive position of manufacturing firms in the South