On the Fucik spectrum of the Laplacian on a torus

We study the Fucik spectrum of the Laplacian on a two-dimensional torus T(2). Exploiting the invariance properties of the domain T(2) with respect to translations we obtain a good description of large parts of the spectrum. In particular, for each eigenvalue of the Laplacian we will find an explicit global curve in the Fucik spectrum which passes through this eigenvalue; these curves are ordered, and we will show that their asymptotic limits are positive. On the other hand, using a topological index based on the mentioned group invariance, we will obtain a variational characterization of global curves in the Fucik spectrum; also these curves emanate from the eigenvalues of the Laplacian, and we will show that they tend asymptotically to zero. Thus, we infer that the variational and the explicit curves cannot coincide globally, and that in fact many curve crossings must occur. We will give a bifurcation result which partially explains these phenomena. (C) 2008 Elsevier Inc. All rights reserved.

The L(2,1)-labelling problem for cubic Cayley graphs on dihedral groups

A k-L(2,1)-labelling of a graph G is a mapping f:V(G)→{0,1,2,…,k} such that |f(u)−f(v)|≥2 if uv∈E(G) and f(u)≠f(v) if u,v are distance two apart. The smallest positive integer k such that G admits a k-L(2,1)-labelling is called the λ-number of G. In this paper we study this quantity for cubic Cayley graphs (other than the prism graphs) on dihedral groups, which are called brick product graphs or honeycomb toroidal graphs. We prove that the λ-number of such a graph is between 5 and 7, and moreover we give a characterisation of such graphs with λ-number 5.

Structure of 2-keto-3-deoxy-6-phosphogluconate (KDPG) aldolase from Pseudomonas putida

2-Keto-3-deoxy-6-phosphogluconate (KDPG) aldolase from Pseudomonas putida is a key enzyme in the Entner-Doudoroff pathway which catalyses the cleavage of KDPG via a class I Schiff-base mechanism. The crystal structure of this enzyme has been refined to a crystallographic residual R = 17.1% (R-free = 21.4%). The N-terminal helix caps one side of the torus of the (betaalpha)(8)-barrel and the active site is located on the opposite, carboxylic side of the barrel. The Schiff-base-forming Lys145 is coordinated by a sulfate (or phosphate) ion and two solvent water molecules. The interactions that stabilize the trimer are predominantly hydrophobic, with the exception of the cyclically permuted bonds formed between Glu132 OE1 of one molecule and Thr129 OG1 of a symmetry-equivalent molecule. Except for the N-terminal helix, the structure of KDPG aldolase from P. putida closely resembles the structure of the homologous enzyme from Escherichia coli.

Free actions of abelian p-groups on the n-torus

In this work we make some contributions to the theory of actions of abelian p-groups on the n-Torus T-n. Set congruent to Z(pk1)(h1) x Z(pk2)(h2) x...x Z(pkr)(hr), r >= 1, k(1) >= k(2) >=...>= k(r) >= 1, p prime. Suppose that the group H acts freely on T-n and the induced representation on pi(1)(T-n) congruent to Z(n) is faithful and has first Betti number b. We show that the numbers n, p, b, k(i) and h(i) (i = 1,..,r) satisfy some relation. In particular, when H congruent to Z(p)(h), the minimum value of n is phi(p) + b when b >= 1. Also when H congruent to Z(pk1) x Z(p) the minimum value of n is phi(p(k1)) + p - 1 + b for b >= 1. Here phi denotes the Euler function.

Morphology and incidence of torus palatinus and mandibularis in Brazilian Indians

The incidence and morphology of torus platinus and mandibularis was verified in 200 Indians, residents of two Brazilian Indian Reserves in Sao Paulo State, Brazil. A low incidence of both types of exostoses was observed, with torus palatinus occurring more frequently than mandibularis. These structures did not occur in individuals less than 10 years of age. Flattened torus palatinus predominated in relation to the other forms.

The Noether theorem for geometric actions and area preserving diffeomorphisms on the torus

We find that within the formalism of coadjoint orbits of the infinite dimensional Lie group the Noether procedure leads, for a special class of transformations, to the constant of motion given by the fundamental group one-cocycle S. Use is made of the simplified formula giving the symplectic action in terms of S and the Maurer-Cartan one-form. The area preserving diffeomorphisms on the torus T2=S1⊗S1 constitute an algebra with central extension, given by the Floratos-Iliopoulos cocycle. We apply our general treatment based on the symplectic analysis of coadjoint orbits of Lie groups to write the symplectic action for this model and study its invariance. We find an interesting abelian symmetry structure of this non-linear problem.

Plasma confinement in tokamaks with robust torus

We present a non-linear symplectic map that describes the alterations of the magnetic field lines inside the tokamak plasma due to the presence of a robust torus (RT) at the plasma edge. This RT prevents the magnetic field lines from reaching the tokamak wall and reduces, in its vicinity, the islands and invariant curve destruction due to resonant perturbations. The map describes the equilibrium magnetic field lines perturbed by resonances created by ergodic magnetic limiters (EMLs). We present the results obtained for twist and non-twist mappings derived for monotonic and non-monotonic plasma current density radial profiles, respectively. Our results indicate that the RT implementation would decrease the field line transport at the tokamak plasma edge. © 2010 Elsevier B.V. All rights reserved.

On singular foliations on the solid torus

We study smooth foliations on the solid torus S1×D2 having S1×{0} and S1×∂D2 as the only compact leaves and S1×{0} as singular set. We show that all other leaves can only be cylinders or planes, and give necessary conditions for the foliation to be a suspension of a diffeomorphism of the disc. © 2013 Elsevier B.V.

A note on pairs of regular foliations on the solid torus

We discuss the geometry of the pair of foliations on a solid torus given by the Reeb foliation together with discs transverse to the boundary of the torus.

Guided Surgery in Unusual Palatal Torus

Palatine torus is a benign congenital outgrowth of bone that affects the hard palate and palatine processes, resulting from the "overworking" of osteoblasts and bone deposition along the line of the palatine fusion. Surgical excision is the only treatment for torus, and such patients are susceptible to intraoperative and postoperative complications of a traumatic, functional, or infectious nature. This article describes an atypical case of torus palatinus measuring 20.31 x 27.25 x 59.20 mm, which is the largest size ever described in the literature. This case required the use of a surgical guide in the intraoperative phase, with viable use in the postoperative phase as well. This guide proved versatile in reducing the risk of undercorrection and complications, offering greater patient comfort.

Clinical, tomographic aspects and relevance of torus palatinus: case report of two sisters

Despite the nomenclature suggested to be a tumor, torus palatinus (TP) is an overgrowth of the bone in the palatal region and represents an anatomic variation. Its prevalence varies among the population studied and its etiology is still unclear; however, it seems to be a multifactorial disorder with genetics and environmental involvement. Surgical removal of the TP is indicated in the following circumstances: (1) deglutition and speech impairment, (2) cancer phobia, (3) traumatized mucosa over the torus, and (4) prosthetic reasons. The aim of this case report is describe cases that occurred in two sisters, emphasizing the genetic etiology of this anatomic variation. In addition, intra-oral exam and computed tomography scan (axial, coronal and sagittal view) provided a detailed assessment of the TP and elimination of other possible diagnoses, furthermore allowed a better analyzes of the anatomic relation with adjacentes structures. No surgical removal was indicated for both cases.

Electrodynamic tether at Jupiter 2. Tour missions after capture

Three separate scenarios of an electrodynamic tether mission at Jupiter following capture of a spacecraft (SC) into an equatorial, highly elliptical orbit around the planet, with perijove at about 1.5 times the Jovian radius, are discussed. Repeated application of Lorentz drag on the spinning tether, at the perijove vicinity, can progressively lower the apojove. One mission involves the tethered-SC rapidly and frequently visiting Galilean moons; elliptical orbits with apojove down at the Ganymede, Europa, and Io orbits are in 2:5, 4:9, and 1:2 resonances with the respective moons. About 20 slow flybys of Io would take place before the accumulated radiation dose exceeds 3 Mrad (Si) at 10 mm Al shield thickness, with a total duration of 5 months after capture (4 months for lowering the apojove to Io and one month for the flybys). The respective number of flybys for Ganymede would be 10 with a total duration of about 9 months. An alternative mission would have the SC acquire a low circular orbit around Jupiter, below the radiation belts, and manoeuvre to get an optimal altitude, with no major radiation effects, in less than 5 months after capture. In a third mission, repeated thrusting at the apojove vicinity, once down at the Io torus, would raise the perijove itself to the torus to acquire a low circular orbit around Io in about 4 months, for a total of 8 months after capture; this corresponds, however, to over 100 apojove passes with an accumulated dose, of about 8.5 Mrad (Si), that poses a critical issue.