Non-uniform Expansions of Real Numbers

We introduce and study non-uniform expansions of real numbers, given by two non-integer bases.

A study case for the analysis of asymptotic expansions beyond the tQSSA for inhibitory mechanisms in enzyme kinetics.

In this paper we study the model of the chemical reaction of fully competitive inhibition and determine the appropriate parameter ∊ (related to the chemical constants of the model), for the application of singular perturbation techniques. We determine the inner and the outer solutions up to the first perturbation order and the uniform expansions. Some numerical results are discussed.

Effect of Amplitude on the Surface Dilational Visco-Elasticity of Protein Solutions

Harmonic drop surface area oscillations are performed at a fixed frequency (0.1 Hz) to measure the dilational visco-elasticity for three proteins: β-casein (BCS), β-lactoglobulin (BLG), and human serum albumin (HSA). The surface area oscillations were performed with different amplitudes in order to find the origin of non-linearity effects. The analysis of data shows that the non-linearity in the equation of state—i.e., the relation between surface pressure and surface concentration of adsorbed protein molecules—is the main source of the amplitude effects on the apparent visco-elasticity, while perturbations due to non-uniform expansions and compressions of the surface layer, inertia effects leading to deviations of the drop profile from the Laplacian shape, or convective transport in the drop bulk are of less importance. While for the globular proteins, HSA and BLG the amplitude effects on the apparent visco-elasticity are rather large, for the non-globular protein BCS this effect is negligible in the studied range of up to 10% area deformation.

Why the high-frequency inverse scattering by topological sensitivity may work

This study deciphers the topological sensitivity (TS) as a tool for the reconstruction and characterization of impenetrable anomalies in the high-frequency regime. It is assumed that the anomaly is simply connected and convex, and that the measurements of the scattered field are of the far-field type. In this setting, the formula for TS—which quantifies the perturbation of a cost functional due to a point-like impenetrable scatterer—is expressed as a pair of nested surface integrals: one taken over the boundary of a hidden obstacle, and the other over the measurement surface. Using multipole expansion, the latter integral is reduced to a set of antilinear forms featuring Green's function and its gradient. The remaining expression is distilled by evaluating the scattered field on the surface of an obstacle via Kirchhoff approximation, and pursuing an asymptotic expansion of the resulting Fourier integral. In this way, the TS is found to survive upon three asymptotic lynchpins, namely (i) the near-boundary approximation for sampling points close to the ‘exposed’ surface of an obstacle; (ii) uniform expansions synthesizing the diffraction catastrophes for sampling points near caustic surfaces, lines and points; and (iii) stationary phase approximation. Within the framework of catastrophe theory, it is shown that, in the case of the full source aperture, the TS is asymptotically dominated by the (explicit) near-boundary term—which explains the previously reported reconstruction capabilities of this class of indicator functionals. The analysis further shows that, when the (Dirichlet or Neumann) character of an anomaly is unknown beforehand, the latter can be effectively exposed by assuming point-like Dirichlet perturbation and considering the sign of the leading term inside the reconstruction.

MRI to delineate the gross tumor volume of nasopharyngeal cancers: which sequences and planes should be used?

Background. Magnetic resonance imaging (MRI) has been found to be better than computed tomography for defining the extent of primary gross tumor volume (GTV) in advanced nasopharyngeal cancer. It is routinely applied for target delineation in planning radiotherapy. However, the specific MRI sequences/planes that should be used are unknown. Methods. Twelve patients with nasopharyngeal cancer underwent primary GTV evaluation with gadolinium-enhanced axial T1 weighted image (T1) and T2 weighted image (T2), coronal T1, and sagittal T1 sequences. Each sequence was registered with the planning computed tomography scans. Planning target volumes (PTVs) were derived by uniform expansions of the GTVs. The volumes encompassed by the various sequences/planes, and the volumes common to all sequences/planes, were compared quantitatively and anatomically to the volume delineated by the commonly used axial T1-based dataset. Results. Addition of the axial T2 sequence increased the axial T1-based GTV by 12% on average (p = 0.004), and composite evaluations that included the coronal T1 and sagittal T1 planes increased the axial T1-based GTVs by 30% on average (p = 0.003). The axial T1-based PTVs were increased by 20% by the additional sequences (p = 0.04). Each sequence/plane added unique volume extensions. The GTVs common to all the T1 planes accounted for 38% of the total volumes of all the T1 planes. Anatomically, addition of the coronal and sagittal-based GTVs extended the axial T1-based GTV caudally and cranially, notably to the base of the skull. Conclusions. Adding MRI planes and sequences to the traditional axial T1 sequence yields significant quantitative and anatomically important extensions of the GTVs and PTVs. For accurate target delineation in nasopharyngeal cancer, we recommend that GTVs be outlined in all MRI sequences/planes and registered with the planning computed tomography scans.

AN INVESTIGATION OF UNIFORM EXPANSIONS OF LARGE ORDER BESSEL FUNCTIONS IN GRAVITATIONAL WAVE SIGNALS

In this work, we extend the analytic treatment of Bessel functions of large order and/or argument. We examine uniform asymptotic Bessel function expansions and show their accuracy and range of validity. Such situations arise in a variety of applications, particularly the Fourier transform (FT) of the gravitational wave (GW) signal from a pulsar, global parameter space correlations of a coherent matched filtering search for continuous GWs from isolated neutron stars and tomographic reconstruction of GW LISA sources. The uniform expansion we consider here is found to be valid in the entire range of the argument.