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.     

An operator splitting approach for the interaction between a fluid and a multilayered poroelastic structure

We develop a loosely coupled fluid‐structure interaction finite element solver based on the Lie operator splitting scheme. The scheme is applied to the interaction between an incompressible, viscous, Newtonian fluid, and a multilayered structure, which consists of a thin elastic layer and a thick poroelastic material. The thin layer is modeled using the linearly elastic Koiter membrane model, while the thick poroelastic layer is modeled as a Biot system. We prove a conditional stability of the scheme and derive error estimates. Theoretical results are supported with numerical examples. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1054–1100, 2015     

Tolerances as images of congruences in varieties defined by linear identities

An identity s = t is linear if each variable occurs at most once in each of the terms s and t. Let T be a tolerance relation of an algebra ${\mathcal{A}}$ in a variety defined by a set of linear identities. We prove that there exist an algebra ${\mathcal{B}}$ in the same variety and a congruence θ of ${\mathcal{B}}$ such that a homomorphism from ${\mathcal{B}}$ onto ${\mathcal{A}}$ maps θ onto T.     

A weak degeneracy revealing decomposition for the CANDECOMP/PARAFAC model

The CANDECOMP/PARAFAC (CP) model is a well known and frequently used tool for extracting substantial information from a three‐way data array. It has several useful characteristics and usually gives meaningful insights about the underlying structure of the data. However, in some cases it has a ‘strange’ behaviour suffering from the so‐called ‘degenerate solutions’, i.e. solutions where the components show a diverging pattern and are meaningless. Several authors have investigated the causes of degeneracy concluding that the phenomenon is due to a lack of minimum of the loss function. In this paper, we study the degeneracy of CP limiting our attention to the two‐component case. The study is done by introducing a canonical form, called 2DR, which is ‘weakly degeneracy revealing’. On the ground of this framework, degeneracy is studied along with some of the remedies proposed in the literature by using a Tucker3 model having a core in the 2DR form. The analysis gives new insights about the behaviour of the CP model and suggests new ideas on how to deal with degeneracy. Copyright © 2010 John Wiley & Sons, Ltd.     

ssDNA binding reveals the atomic structure of graphene

We used AFM to investigate the interaction of polyelectrolytes such as ssDNA and dsDNA molecules with graphene as a substrate. Graphene is an appropriate substrate due to its planarity, relatively large surfaces that are detectable via an optical microscope, and straightforward identification of the number of layers. We observe that in the absence of the screening ions deposited ssDNA will bind only to the graphene and not to the SiO(2) substrate, confirming that the binding energy is mainly due to the π-π stacking interaction. Furthermore, deposited ssDNA will map the graphene underlying structure. We also quantify the π-π stacking interaction by correlating the amount of deposited DNA with the graphene layer thickness. Our findings agree with reported electrostatic force microscopy (EFM) measurements. Finally, we inspected the suitability of using a graphene as a substrate for DNA origami-based nanostructures.