A note on Sommerfeld's diffraction problem

A direct transform technique is found to be most suitable for attacking two-dimensional diffraction problems. As a first example of the application of the technique, the well-known Sommerfeld problem is reconsidered and the solution of the problem of diffraction, by a half-plane, of a cylindrical pulse is made use of in deducing the solution of the problem of diffraction of a plane wave by a soft half-plane. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.

A note on fractal dimensions of biomedical waveforms

In this paper, we study performance of Katz method of computing fractal dimension of waveforms, and its estimation accuracy is compared with Higuchi's method. The study is performed on four synthetic parametric fractal waveforms for which true fractal dimensions can be calculated, and real sleep electroencephalogram. The dependence of Katz's fractal dimension on amplitude, frequency and sampling frequency of waveforms is noted. Even though the Higuchi's method has given more accurate estimation of fractal dimensions, the study suggests that the results of Katz's based fractal dimension analysis of biomedical waveforms have to be carefully interpreted.

A note on the Hadwiger number of circular arc graphs

The intention of this note is to motivate the researchers to study Hadwiger's conjecture for circular arc graphs. Let η(G) denote the largest clique minor of a graph G, and let χ(G) denote its chromatic number. Hadwiger's conjecture states that η(G)greater-or-equal, slantedχ(G) and is one of the most important and difficult open problems in graph theory. From the point of view of researchers who are sceptical of the validity of the conjecture, it is interesting to study the conjecture for graph classes where η(G) is guaranteed not to grow too fast with respect to χ(G), since such classes of graphs are indeed a reasonable place to look for possible counterexamples. We show that in any circular arc graph G, η(G)less-than-or-equals, slant2χ(G)−1, and there is a family with equality. So, it makes sense to study Hadwiger's conjecture for this family.

Note: Creep behaviour of high density polyethylene

The Ramberg-Osgood type constitutive law for creep suggested by Iyengar2 has been verified on high-density polyethylene. The time functions are evaluated from the experimental data of Scheweiker and Sidebottom3. It is found that the creep behaviour of the above material can be represented by the Ramberg-Osgood type law.

Note on the locally product and almost locally product structures

This paper is concerned with a study of some of the properties of locally product and almost locally product structures on a differentiable manifold X n of class C k . Every locally product space has certain almost locally product structures which transform the local tangent space to X n at an arbitrary point P in a set fashion: this is studied in Theorem (2.2). Theorem (2.3) considers the nature of transformations that exist between two co-ordinate systems at a point whenever an almost locally product structure has the same local representation in each of these co-ordinate systems. A necessary and sufficient condition for X n to be a locally product manifold is obtained in terms of the pseudo-group of co-ordinate transformations on X n and the subpseudo-groups [cf., Theoren (2.1)]. Section 3 is entirely devoted to the study of integrable almost locally product structures.

A note on the measurement of travel times using bender and extender elements

By using the bender and extender elements tests, the travel times of the shear (S) and the primary (P) waves were measured for dry sand samples at different relative densities and effective confining pressures. Three methods of interpretations, namely, (i) the first time of arrival, (ii) the first peak to peak, and (iii) the cross-correlation method, were employed. All the methods provide almost a unique answer associated with the P-wave measurements. On contrary, a difference was noted in the arrival times obtained from the different methods for the S-wave due to the near field effect. The resonant column tests in the torsional mode were also performed to check indirectly the travel time of the shear wave. The study reveals that as compared to the S-wave, it is more reliable to depend on the arrival times' measurement for the P-wave. (C) 2010 Elsevier Ltd. All rights reserved.

A note on the measurement of travel times using bender and extender elements

By using the bender and extender elements tests, the travel times of the shear (S) and the primary (P) waves were measured for dry sand samples at different relative densities and effective confining pressures. Three methods of interpretations, namely, (i) the first time of arrival, (ii) the first peak to peak, and (iii) the cross-correlation method, were employed. All the methods provide almost a unique answer associated with the P-wave measurements. On contrary, a difference was noted in the arrival times obtained from the different methods for the S-wave due to the near field effect. The resonant column tests in the torsional mode were also performed to check indirectly the travel time of the shear wave. The study reveals that as compared to the S-wave, it is more reliable to depend on the arrival times’ measurement for the P-wave.

Note: Dynamic point spread function for single and multiphoton fluorescence microscopy

We propose and demonstrate a dynamic point spread function (PSF) for single and multiphoton fluorescence microscopy. The goal is to generate a PSF whose shape and size can be maneuvered from highly localized to elongated one, thereby allowing shallow-to-depth excitation capability during active imaging. The PSF is obtained by utilizing specially designed spatial filter and dynamically altering the filter parameters. We predict potential applications in nanobioimaging and fluorescence microscopy.

Note: Dynamic point spread function for single and multiphoton fluorescence microscopy

We propose and demonstrate a dynamic point spread function (PSF) for single and multiphoton fluorescence microscopy. The goal is to generate a PSF whose shape and size can be maneuvered from highly localized to elongated one, thereby allowing shallow-to-depth excitation capability during active imaging. The PSF is obtained by utilizing specially designed spatial filter and dynamically altering the filter parameters. We predict potential applications in nanobioimaging and fluorescence microscopy.

A Note on Diffraction of Water Waves by a Nearly Vertical Barrier

A simplified perturbational analysis is employed, together with the application of Green's theorem, to determine the first-order corrections to the reflection and transmission coefficients in the problem of diffraction of surface water waves by a nearly vertical barrier in two basically important cases: (i) when the barrier is partially immersed and (ii) when the barrier is completely submerged. The present analysis produces the desired results fairly easily and relatively quickly as compared with the known integral equation approach to this class of diffraction problems.

A note on the porous-wavemaker problem

A complete analytical solution is obtained, by using an integral transform method, for the porous-wavemaker problem, when the effect of surface tension is taken into account on the free surface of water of finite-depth in which surface waves are produced by small horizontal oscillations of a porous vertical plate. The final results are expressed in the form of convergent integrals as well as series and known results are reproduced when surface tension is neglected.

A note on the Berry phase for systems having one degree of freedom

A one-dimensional arbitrary system with quantum Hamiltonian H(q, p) is shown to acquire the 'geometric' phase gamma (C)=(1/2) contour integral c(Podqo-qodpo) under adiabatic transport q to q+q+qo(t) and p to p+po(t) along a closed circuit C in the parameter space (qo(t), po(t)). The non-vanishing nature of this phase, despite only one degree of freedom (q), is due ultimately to the underlying non-Abelian Weyl group. A physical realisation in which this Berry phase results in a line spread is briefly discussed.