On the Residues of Binomial Coefficients and Their Products Modulo Prime Powers

In this paper, we show several arithmetic properties on the residues of binomial coefficients and their products modulo prime powers, e.g., for any distinct odd primes p and q. Meanwhile, we discuss the connections with the prime recognitions. Received November 5, 1998, Accepted December 7, 2000.     

Detection of incipient gear failures using statistical techniques

Gears are important components in most power transmission mechanisms.Failures of gears can cause heavy losses in industry. Conditionmonitoring and fault diagnosis of gears is therefore importantto improve safety and reliability of gearbox operations andreduce losses caused by gear failures. This research proposesa new diagnostic approach based on the statistical analysisof data. It investigates the use of Principal Components Analysis(PCA) to detect growing local faults in a two-stage industrialhelical gearbox. In this research, the vibration signal is usedto monitor fault conditions and a broken tooth is simulatedas a local fault. Since the early detection of faults is a challenge,small fault conditions were tested first as well as severe faultconditions. In order to examine the ability of the PCA to detectfault conditions, first the PCA-based model was created fornormal operating conditions. Any unexpected event such as afault condition causes a significant deviation from the PCAmodel, which is obtained from the normal condition data of thegearbox. The Square Prediction Error (SPE) was calculated todetect the fault conditions. When the vibration signal fromthe gearbox is representative of normal operation, the valueof the SPE shows very little fluctuation and remains under acertain threshold value. However, in the presence of the faultthe SPE fluctuates considerably beyond the threshold value.It is shown that the PCA-based statistical approach cannot onlybe used to detect severe fault conditions, but that it alsoreveals small growing fault conditions at very early stage.The technique also provides information about the state of thefault such as the location of the fault as well as its severity. Received 5 March 2001. Revised 12 December 2001. Accepted 17 January 2002.     

NMR chemical shifts in amino acids: effects of environments in the condensed phase

We present calculations of NMR chemical shifts in crystalline phases of some representative amino acids such as glycine, alanine, and alanyl-alanine. We explore the effects of environment on the chemical shifts in selected glycine geometries ranging from the crystalline phase to completely isolated molecules. In the crystalline and dilute molecular limits, the calculated distinct NMR chemical shifts are attributed to intermolecular hydrogen-bonds and dipole electric field effects, respectively.     

Potential flow around obstacles using the scaled boundary finite‐element method

The scaled boundary finite‐element method is a novel semi‐analytical technique, combining the advantages of the finite element and the boundary element methods with unique properties of its own. The method works by weakening the governing differential equations in one co‐ordinate direction through the introduction of shape functions, then solving the weakened equations analytically in the other (radial) co‐ordinate direction. These co‐ordinate directions are defined by the geometry of the domain and a scaling centre. The method can be employed for both bounded and unbounded domains. This paper applies the method to problems of potential flow around streamlined and bluff obstacles in an infinite domain. The method is derived using a weighted residual approach and extended to include the necessary velocity boundary conditions at infinity. The ability of the method to model unbounded problems is demonstrated, together with its ability to model singular points in the near field in the case of bluff obstacles. Flow fields around circular and square cylinders are computed, graphically illustrating the accuracy of the technique, and two further practical examples are also presented. Comparisons are made with boundary element and finite difference solutions. Copyright © 2003 John Wiley & Sons, Ltd.     

Critical phenomena and relativistic gravitational collapse

Recent calculations have shown the existence of critical phenomena in general relativity associated with the collapse of wavepackets of massless fields that are near, in parameter space, the onset of black hole formation (the critical point). Two physically distinct systems have been explored: collapse of spherically-symmetric massless scalar field and collapse of vacuum, axisymmetric gravitational waves. Nonlinear effects dominate near the critical point. Black-hole mass serves as an order parameter and has a power-law dependence on critical separation in the supercritical region of parameter space. Remarkably, the values of the critical exponent of the power law are nearly identical in the two systems. The nonlinearity induces the fields to oscillate. Each successive oscillation is an echo, obeying a spatial and temporal scaling relation.     

Canp-adic numbers be useful to regularize divergent expectation values of quantum observables?

We show howp-adic analysis can be used in some cases to treat divergent series in quantum mechanics. We consider examples in which the usual theory of the Schrödinger equation would give rise to an infinite expectation value of the energy operator. By usingp-adic analysis, we are able to get a convergent expansion and obtain a finite rational value for the energy. We present also the main ideas to interpret a quantum mechanical state by means ofp-adic statistics.     

The definition and calculation of interfacial energies for thin films

We provide a new definition of the interfacial energy which eliminates three physically extraneous contributions from the conventional definition: (1) the strain or stress energy due to lattice mismatch between film and substrate; (2) the surface energy of the film-vacuum interface; and, (3) the substrate surface energy contribution from substrate layers below the film layers. This new interface energy then quantifies the variation in interactions among film/substrate, film/film and substrate/substrate bonding. Using this new definition, we derive the equations for evaluation of the interfacial energy in terms of the interaction energy for any atom in each layer of the film/substrate, film/film and substrate/substrate systems. With this formulation, it is simple to determine the dependence of the interfacial energy on the film thickness using virtually any interaction potential. Using a corrected effective medium theory, we present results for a few pseudomorphic film systems containing Ni/Cu, Ni/Ag, Cu/Ag and Rh/Ag on (111) and (100) surfaces. These systems cover a wide range of lattice mismatch and alloy formation energies. The results demonstrate that the new definition of interfacial energy converges after only 3–4 film layers, regardless of the degree of lattice mismatch. We also show that the interfacial energies at (100) and (111) interfaces differ and that the interfacial energy for a given pair of materials depends on which of the materials is the film.     

The dynamical behavior of the discontinuous Galerkin method and related difference schemes

We study the dynamical behavior of the discontinuous Galerkin finite element method for initial value problems in ordinary differential equations. We make two different assumptions which guarantee that the continuous problem defines a dissipative dynamical system. We show that, under certain conditions, the discontinuous Galerkin approximation also defines a dissipative dynamical system and we study the approximation properties of the associated discrete dynamical system. We also study the behavior of difference schemes obtained by applying a quadrature formula to the integrals defining the discontinuous Galerkin approximation and construct two kinds of discrete finite element approximations that share the dissipativity properties of the original method.