Sharp Sobolev inequality involving a critical nonlinearity on a boundary

We consider the solvability of the Neumann problem for the equation -Delta u + lambda u = 0, partial derivative u/partial derivative v = Q(x)vertical bar u vertical bar(q-2)u on partial derivative Omega, where Q is a positive and continuous coefficient on partial derivative Omega, lambda is a parameter and q = 2(N - 1)/(N - 2) is a critical Sobolev exponent for the trace embedding of H-1(Omega) into L-q(partial derivative Omega). We investigate the joint effect of the mean curvature of partial derivative Omega and the shape of the graph of Q on the existence of solutions. As a by product we establish a sharp Sobolev inequality for the trace embedding. In Section 6 we establish the existence of solutions when a parameter lambda interferes with the spectrum of -Delta with the Neumann boundary conditions. We apply a min-max principle based on the topological linking.

Some twisted results

The Drinfeld twist for the opposite quasi-Hopf algebra, H-COP, is determined and is shown to be related to the (second) Drinfeld twist on a quasi-Hopf algebra. The twisted form of the Drinfeld twist is investigated. In the quasi-triangular case, it is shown that the Drinfeld u-operator arises from the equivalence of H-COP to the quasi-Hopf algebra induced by twisting H with the R-matrix. The Altschuler-Coste u-operator arises in a similar way and is shown to be closely related to the Drinfeld u-operator. The quasi-cocycle condition is introduced and is shown to play a central role in the uniqueness of twisted structures on quasi-Hopf algebras. A generalization of the dynamical quantum Yang-Baxter equation, called the quasi-dynamical quantum Yang-Baxter equation, is introduced.

On the chaotic rotation of a liquid-filled gyrostat via the Melnikov-Holmes-Marsden integral

The non-linear motions of a gyrostat with an axisymmetrical, fluid-filled cavity are investigated. The cavity is considered to be completely filled with an ideal incompressible liquid performing uniform rotational motion. Helmholtz theorem, Euler's angular momentum theorem and Poisson equations are used to develop the disturbed Hamiltonian equations of the motions of the liquid-filled gyrostat subjected to small perturbing moments. The equations are established in terms of a set of canonical variables comprised of Euler angles and the conjugate angular momenta in order to facilitate the application of the Melnikov-Holmes-Marsden (MHM) method to investigate homoclinic/heteroclinic transversal intersections. In such a way, a criterion for the onset of chaotic oscillations is formulated for liquid-filled gyrostats with ellipsoidal and torus-shaped cavities and the results are confirmed via numerical simulations. (c) 2006 Elsevier Ltd. All rights reserved.

Mechanisms involved in the swelling of erythrocytes caused by Pacific and Caribbean ciguatoxins

The mechanisms underlying the swelling of frog red blood cells (RBC), induced by Pacific (P-CTX-1) and Caribbean (C-CTX-1) ciguatoxins (CTXs), were investigated by measuring the length, width and surface of their elliptic shape. P-CTX-1 (0.5 to 5 nM) and C-CTX-1 (1 mu M) induced RBC swelling within 60 min. The CTXs-induced RBC swelling was blocked by apamin (1 mu M) and by Sr2+ (1 mu M). P-CTX-1-induced RBC swelling was prevented and inhibited by H-[1,2,4]oxadiazolo[4,3-a]quinoxalin-1-one(27 mu M), an inhibitor Of Soluble guanylate cyclase (sGC), and NOS blockade by NG methyl-L-arginine (L-NMA; 10 mu M). Cytochalasin D (cytD, 10 mu M) increased RBC surface and mimicked CTX effect but did not prevent the P-CTX-1-induced L-NMA-sensitive extra increase. Calculations revealed that P-CTX-1 and cytD increase RBC total surface envelop and volume. These data strongly suggest that the molecular mechanisms underlying CTXs-induced RBC swelling involve the NO pathway by an activation of the inducible NOS, leading to sGC activation which modulates intracellular cGMP and regulates L-type Ca2+ channels. The resulting increase in intracellular Ca2+ content, in turn, disrupts the actin cytoskeleton, which causes a water influx and triggers a Ca2+-activated K+ current through SK2 isoform channels. (c) 2005 Elsevier Inc. All rights reserved.

Sigmoidal cosine series on the interval

We construct a set of functions, say, psi([r])(n) composed of a cosine function and a sigmoidal transformation gamma(r) of order r > 0. The present functions are orthonormal with respect to a proper weight function on the interval [-1, 1]. It is proven that if a function f is continuous and piecewise smooth on [-1, 1] then its series expansion based on psi([r])(n) converges uniformly to f so long as the order of the sigmoidal transformation employed is 0 < r

The 2df SDSS LRG and QSO survey: evolution of the luminosity function of luminous red galaxies to z=0.6

We present new measurements of the luminosity function (LF) of luminous red galaxies (LRGs) from the Sloan Digital Sky Survey (SDSS) and the 2dF SDSS LRG and Quasar (2SLAQ) survey. We have carefully quantified, and corrected for, uncertainties in the K and evolutionary corrections, differences in the colour selection methods, and the effects of photometric errors, thus ensuring we are studying the same galaxy population in both surveys. Using a limited subset of 6326 SDSS LRGs (with 0.17 < z < 0.24) and 1725 2SLAQ LRGs (with 0.5 < z < 0.6), for which the matching colour selection is most reliable, we find no evidence for any additional evolution in the LRG LF, over this redshift range, beyond that expected from a simple passive evolution model. This lack of additional evolution is quantified using the comoving luminosity density of SDSS and 2SLAQ LRGs, brighter than M-0.2r - 5 log h(0.7) = - 22.5, which are 2.51 +/- 0.03 x 10(-7) L circle dot Mpc(-3) and 2.44 +/- 0.15 x 10(-7) L circle dot Mpc(-3), respectively (< 10 per cent uncertainty). We compare our LFs to the COMBO-17 data and find excellent agreement over the same redshift range. Together, these surveys show no evidence for additional evolution (beyond passive) in the LF of LRGs brighter than M-0.2r - 5 log h(0.7) = - 21 ( or brighter than similar to L-*).. We test our SDSS and 2SLAQ LFs against a simple 'dry merger' model for the evolution of massive red galaxies and find that at least half of the LRGs at z similar or equal to 0.2 must already have been well assembled (with more than half their stellar mass) by z similar or equal to 0.6. This limit is barely consistent with recent results from semi-analytical models of galaxy evolution.

The 2dF-SDSS LRG and QSO survey: the star formation histories of luminous red galaxies

We present a detailed investigation into the recent star formation histories of 5697 luminous red galaxies (LRGs) based on the H delta (4101 angstrom), and [O II] (3727 angstrom) lines and the D4000 index. LRGs are luminous (L > 3L*) galaxies which have been selected to have photometric properties consistent with an old, passively evolving stellar population. For this study, we utilize LRGs from the recently completed 2dF-SDSS LRG and QSO Survey (2SLAQ). Equivalent widths of the H delta and [O II] lines are measured and used to define three spectral types, those with only strong H delta absorption (k+a), those with strong [O II] in emission (em) and those with both (em+a). All other LRGs are considered to have passive star formation histories. The vast majority of LRGs are found to be passive (similar to 80 per cent); however, significant numbers of k+a (2.7 per cent), em+a (1.2 per cent) and em LRGs (8.6 per cent) are identified. An investigation into the redshift dependence of the fractions is also performed. A sample of SDSS MAIN galaxies with colours and luminosities consistent with the 2SLAQ LRGs is selected to provide a low-redshift comparison. While the em and em+a fractions are consistent with the low-redshift SDSS sample, the fraction of k+a LRGs is found to increase significantly with redshift. This result is interpreted as an indication of an increasing amount of recent star formation activity in LRGs with redshift. By considering the expected lifetime of the k+a phase, the number of LRGs which will undergo a k+a phase can be estimated. A crude comparison of this estimate with the predictions from semi-analytic models of galaxy formation shows that the predicted level of k+a and em+a activities is not sufficient to reconcile the predicted mass growth for massive early types in a hierarchical merging scenario.

Cracks of higher modes in Cosserat continua

Rotational degrees of freedom in Cosserat continua give rise to higher fracture modes. Three new fracture modes correspond to the cracks that are surfaces of discontinuities in the corresponding components of independent Cosserat rotations. We develop a generalisation of J- integral that includes these additional degrees of freedom. The obtained path-independent integrals are used to develop a criterion of crack propagation for a special type of failure in layered materials with sliding layers. This fracture propagates as a progressive bending failure of layers – a “bending crack that is, a crack that can be represented as a distribution of discontinuities in the layer bending. This situation is analysed using a 2D Cosserat continuum model. Semi-infinite bending crack normal to layering is considered. The moment stress concentrates along the line that is a continuation of the crack and has a singularity of the power − 1/4. A model of process zone is proposed for the case when the breakage of layers in the process of bending crack propagation is caused by a crack (microcrack in our description) growing across the layer adjacent to the crack tip. This growth is unstable (in the moment-controlled loading), which results in a typical descending branch of moment stress – rotation discontinuity relationship and hence in emergence of a Barenblatt-type process zone at the tip of the bending crack.