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.     

Experimental behaviour of the free surface in a uniform laminar stream

Summary It has been experimentally shown that in a uniform laminar free surface flow, the boundary conditions at the free surface cannot be represented by a simple non-shear condition.This statement is in particular supported by some experiments performed by the author on a high viscosity oil flow in a rectangular pipe. A surface shear viscosity may be introduced in order to characterize the behaviour of the free surface (Boussinesq).The velocity profile was measured along the axis of simmetry of the cross section. The results were in agreement with these of some numerical computations based on the above model. Similar agreement was not achieved on comparing the experimental and theoretical values obtained for the shape factor, a global parameter depending on the shape of the cross section.Such a disagreement cannot be due to experimental error and suggests that the above model is unable.

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Sommario E' fatto sperimentalmente accertato che in un moto laminare uniforme le condizioni al contorno sulla superficie libera non possono essere rappresentate dalla semplice assenza di tensioni tangenziali superficiali. Una chiara conferma viene dalla attuale serie di misure eseguite su una condotta rettangolare percorsa da olio.Per caratterizzare il comportamento della superficie libera si introduce un coefficiente di viscosità tangenziale superficiale, nell'ipotesi di comportamento newtoniano della superficie stessa (Boussinesq).I confronti fra i nostri risultati sperimentali ed i risultati numerici derivanti da tale ipotesi mostrano una possibile concordanza nella distribuzione di velocità sull'asse di simmetria della condotta ma mettono in evidenza andamenti decisamente discordanti per quanto riguarda il comportamento di un coefficiente globale, variabile con la forma della sezione.Tale discordanza non è attribuibile ad errori di misura ed indica l'inadeguatezza di un modello basato sull'ipotesi di comportamento newtoniano della superficie libera.

     

A mathematical model of anorexia and bulimia

In this paper, we propose a mathematical model to study the dynamics of anorexic and bulimic populations. The model proposed takes into account, among other things, the effects of peers' influence, media influence, and education. We prove the existence of three possible equilibria that without media influences are disease‐free, bulimic‐endemic, and endemic. Neglecting media and education effects, we investigate the stability of such equilibria, and we prove that under the influence of media, only one of such equilibria persists and becomes a global attractor. Which of the three equilibria becomes global attractor depends on the other parameters. Copyright © 2014 John Wiley & Sons, Ltd.