Abstract:
The resonances for the Dirichlet and Neumann Laplacian are studied on compactly perturbed waveguides. An upper bound on the number of resonances near the physical plane is proven. In the absence of resonances, an upper bound is proven for the localised resolvent. This is then used to prove that the existence of a quasimode whose asymptotics is bounded away from the thresholds implies the existence of resonances converging to the real axis.

Abstract:
The Laplace operator is considered for waveguides perturbed by a periodic structure consisting of N congruent obstacles spanning the waveguide. Neumann boundary conditions are imposed on the periodic structure, and either Neumann or Dirichlet conditions on the guide walls. It is proven that there are at least N (resp. N-1) trapped modes in the Neumann case (resp. Dirichlet case) under fairly general hypotheses, including the special case where the obstacles consist of line segments placed parallel to the waveguide walls. This work should be viewed as an extension of "Periodic structures on waveguides" by Linton and McIvor.

Abstract:
Barbour, Foster and \'{O} Murchadha have recently developed a new framework, called here {\it{the 3-space approach}}, for the formulation of classical bosonic dynamics. Neither time nor a locally Minkowskian structure of spacetime are presupposed. Both arise as emergent features of the world from geodesic-type dynamics on a space of 3-dimensional metric--matter configurations. In fact gravity, the universal light cone and Abelian gauge theory minimally coupled to gravity all arise naturally through a single common mechanism. It yields relativity -- and more -- without presupposing relativity. This paper completes the recovery of the presently known bosonic sector within the 3-space approach. We show, for a rather general ansatz, that 3-vector fields can interact among themselves only as Yang--Mills fields minimally coupled to gravity.

Abstract:
The human cochlea is a remarkable device, able to discern extremely small amplitude sound pressure waves, and discriminate between very close frequencies. Simulation of the cochlea is computationally challenging due to its complex geometry, intricate construction and small physical size. We have developed, and are continuing to refine, a detailed three-dimensional computational model based on an accurate cochlear geometry obtained from physical measurements. In the model, the immersed boundary method is used to calculate the fluid-structure interactions produced in response to incoming sound waves. The model includes a detailed and realistic description of the various elastic structures present. In this paper, we describe the computational model and its performance on the latest generation of shared memory servers from Hewlett Packard. Using compiler generated threads and OpenMP directives, we have achieved a high degree of parallelism in the executable, which has made possible several large scale numerical simulation experiments that study the interesting features of the cochlear system. We show several results from these simulations, reproducing some of the basic known characteristics of cochlear mechanics.

Abstract:
We consider a number of boundary value problems involving the $p$-Laplacian. The model case is $-\Delta_p u=V|u|^{p-2}u$ for $u\in W_0^{1,2}(D)$ with $D$ a bounded domain in ${\bf R}^n$. We derive necessary conditions for the existence of nontrivial solutions. These conditions usually involve a lower bound for a product of powers of the norm of $V$, the measure of $D$, and a sharp Sobolev constant. In most cases, these inequalities are best possible. Applications to non-linear eigenvalue problems are also discussed.

Abstract:
We describe several techniques for using bulk matter for special purpose computation. In each case it is necessary to use an evolutionary algorithm to program the substrate on which the computation is to take place. In addition, the computation comes about as a result of nearest neighbour interactions at the nano- micro- and meso-scale. In our first example we describe evolving a saw-tooth oscillator in a CMOS substrate. In the second example we demonstrate the evolution of a tone discriminator by exploiting the physics of liquid crystals. In the third example we outline using a simulated magnetic quantum dot array and an evolutionary algorithm to develop a pattern matching circuit. Another example we describe exploits the micro-scale physics of charge density waves in crystal lattices. We show that vastly different resistance values can be achieved and controlled in local regions to essentially construct a programmable array of coupled micro-scale quasiperiodic oscillators. Lastly we show an example where evolutionary algorithms could be used to control density modulations, and therefore refractive index modulations, in a fluid for optical computing.

Abstract:
(abridged) Based on observations of the Seyfert nucleus in NGC1068 with ASCA, RXTE and BeppoSAX, we report the discovery of a flare (increase in flux by a factor of ~1.6) in the 6.7 keV Fe K line component between observations obtained 4 months apart, with no significant change in the other (6.21, 6.4, and 6.97 keV) Fe K_alpha line components. During this time, the continuum flux decreased by ~20%. The RXTE spectrum requires an Fe K absorption edge near 8.6 keV (Fe XXIII - XXV). The spectral data indicate that the 2-10 keV continuum emission is dominated (~2/3 of the luminosity) by reflection from a previously unidentified region of warm, ionized gas located <~ 0.2 pc from the AGN. The remaining ~1/3 of the observed X-ray emission is reflected from optically thick, neutral gas. The inferred properties of the warm reflector (WR) are: size (diameter) <~0.2 pc, gas density n >~ 10^{5.5} /cm3, ionization parameter xi approx 10^{3.5} erg cm/s, and covering fraction 0.003 (L_0/10^{43.5} erg/s)^{-1} < (Omega/4 pi) < 0.024 (L_0/10^{43.5})^{-1}, where L_0 is the intrinsic 2-10 keV X-ray luminosity of the AGN. We suggest that the WR gas is the source of the (variable) 6.7 keV Fe line emission, and the 6.97 keV Fe line emission. The 6.7 keV line flare is assumed to be due to an increase in the emissivity of the WR gas from a decrease (by 20-30%) in L_0. The properties of the WR are most consistent with an intrinsically X-ray weak AGN with L_0 approx 10^{43.0} erg/s. The optical and UV emission that scatters from the WR into our line of sight is required to suffer strong extinction, which can be reconciled if the line-of-sight skims the outer surface of the torus. Thermal bremsstrahlung radio emission from the WR may be detectable in VLBA radio maps of the NGC 1068 nucleus.

Abstract:
We present a scale-invariant theory, conformal gravity, which closely resembles the geometrodynamical formulation of general relativity (GR). While previous attempts to create scale-invariant theories of gravity have been based on Weyl's idea of a compensating field, our direct approach dispenses with this and is built by extension of the method of best matching w.r.t scaling developed in the parallel particle dynamics paper by one of the authors. In spatially-compact GR, there is an infinity of degrees of freedom that describe the shape of 3-space which interact with a single volume degree of freedom. In conformal gravity, the shape degrees of freedom remain, but the volume is no longer a dynamical variable. Further theories and formulations related to GR and conformal gravity are presented. Conformal gravity is successfully coupled to scalars and the gauge fields of nature. It should describe the solar system observations as well as GR does, but its cosmology and quantization will be completely different.

Abstract:
We consider a number of linear and non-linear boundary value problems involving generalized Schr\"odinger equations. The model case is $-\Delta u=Vu$ for $u\in W_0^{1,2}(D)$ with $D$ a bounded domain in ${\bf R^n}$. We use the Sobolev embedding theorem, and in some cases the Moser-Trudinger inequality and the Hardy-Sobolev inequality, to derive necessary conditions for the existence of nontrivial solutions. These conditions usually involve a lower bound for a product of powers of the norm of $V$, the measure of $D$, and a sharp Sobolev constant. In most cases, these inequalities are best possible.

Abstract:
Blood samples from 49 patients with metastatic breast cancer were processed using the CellSearch？ system (Veridex, LLC, Raritan, NJ, USA), in parallel with our CTC assay method. We used anti-CK alone or in combination with anti-EpCAM antibodies for CTC enrichment. Brightfield and fluorescence labeled anti-CK, anti-CD45 and DAPI (nuclear stain) images were used for CTC identification. The Ariol？ system (Genetix USA Inc, San Jose, CA, USA) was used for automated cell image capture and analysis of CTCs on glass slides.Our method has the capability to enrich three types of CTCs including CK+&EpCAM+, CK+&EpCAM-/low, and CK-/low&EpCAM+ cells. In the blind method comparison, our anti-CK antibody enrichment method showed a significantly higher CTC positive rate (49% vs. 29%) and a larger dynamic CTC detected range (1 to 571 vs. 1 to 270) than that of the CellSearch？ system in the total of 49 breast cancer patients. Our method detected 15 to 111% more CTCs than the CellSearch？ method in patients with higher CTC counts (>20 CTCs per 7.5 ml of blood). The three fluorescent and brightfield images from the Ariol？ system reduced the number of false-positive CTC events according to the established CTC criteria.Our data indicate that the tumor-specific intracellular CK marker could be used for efficient CTC enrichment. Enrichment with anti-CK alone or combined with anti-EpCAM antibodies significantly enhances assay sensitivity. The three fluorescent and brightfield superior images with the Ariol？ system reduced false-positive CTC events.Circulating tumor cells (CTCs) are detectable in most blood samples from patients with metastatic cancer using different technologies. CTCs are rare and need to be enriched from the patients' blood sample for better detection [1,2]. CTC analysis has been performed mostly in breast cancer, the second leading cause of cancer death in women in the US and the Western world. Metastatic breast cancer occurs when tumor cells grow unregulated and eventually