Bornological Convergences on the Hyperspace of a Uniformizable Space

Bornological convergence is a generalization of the well known Attouch–Wets convergence. Our aim is to compare lower, upper and “two-sided” convergences generated by two compatible uniformities and two arbitrary bornologies. Moreover, the comparison of convergences induced by bounded-proximal topologies is characterized.     

Step bunching during the epitaxial growth of a generic binary-compound thin film

A discrete-continuum theory for the step-flow growth by chemical beam or metalorganic vapor-phase epitaxy of a generic binary-compound thin film is developed from basic considerations of continuum physics in accordance with the second law of thermodynamics. Our theory accounts for dissipation, chemical and otherwise; allows for departures from equilibrium; and generalizes the classical, variationally derived Gibbs-Thomson relation along the steps. In contrast to existing models, the diffusing species are coupled through a chemical reaction whereby bulk molecules are crystallized from adatoms attaching to the step edges. The linear stability analysis of the resulting free-boundary problem for a periodic train of rectilinear steps yields pairing in the presence of the normal Ehrlich-Schwoebel barrier for both species, counter to the predictions of standard Burton-Cabrera-Frank models for single-species growth. In particular, we show that the onset of step bunching occurs as long as the adatom equilibrium coverage of either species is sufficiently high, a condition met, e.g., during the epitaxy of gallium arsenide. The physical origin of this instability is to be found in the dependence of the step chemical potential on the jump in the adatom grand canonical potential, a term that couples adjacent terraces and—counter to elastic, entropic, or electrostatic interactions between steps—is attractive.     

Diffusion-type operators,Liouville theorems and gradient estimates on complete manifolds

We study Liouville theorems and gradient estimates for solutions of Eq. (1.1) with the help of a diffusion operator and the related geometry.     

Path-integral solution of the one-dimensional Dirac quantum cellular automaton

Quantum cellular automata, which describe the discrete and exactly causal unitary evolution of a lattice of quantum systems, have been recently considered as a fundamental approach to quantum field theory and a linear automaton for the Dirac equation in one dimension has been derived. In the linear case a quantum cellular automaton is isomorphic to a quantum walk and its evolution is conveniently formulated in terms of transition matrices. The semigroup structure of the matrices leads to a new kind of discrete path-integral, different from the well known Feynman checkerboard one, that is solved analytically in terms of Jacobi polynomials of the arbitrary mass parameter.     

On the number of minimal models of a log smooth threefold

We give a topological bound on the number of minimal models of a class of three-dimensional log smooth pairs of log general type.     

Hypertractions and hyperstresses convey the same mechanical information

A strengthened and generalized version of the standard Virtual Work Principle is shown to imply, in addition to bulk and boundary balances, a one-to-one correspondence between surface and edge hypertractions and hyperstress fields in second-gradient continua. When edge hypertractions are constitutively taken null, the hyperstress is shown to take the form it has for a relevant example of second-gradient fluid-like material, referred to as a Navier–Stokes—α fluid.     

The adjoint representation inside the exterior algebra of a simple Lie algebra

For a simple complex Lie algebra g$\mathfrak{g}$ we study the space of invariants A=(?g^?⊗g^?)^g$A={(?{\mathfrak{g}}^{?}\otimes {\mathfrak{g}}^{?})}^{\mathfrak{g}}$, which describes the isotypic component of type g$\mathfrak{g}$ in ?g^?$?{\mathfrak{g}}^{?}$, as a module over the algebra of invariants (?g^?)^g${(?{\mathfrak{g}}^{?})}^{\mathfrak{g}}$. As main result we prove that A   is a free module, of rank twice the rank of g$\mathfrak{g}$, over the exterior algebra generated by all primitive invariants in (?g^?)^g${(?{\mathfrak{g}}^{?})}^{\mathfrak{g}}$, with the exception of the one of highest degree.