Single Discontinuity Research Articles

Waves Past Porous Structures in a Two-layer Fluid

Havelock’s type of expansion theorems, for an integrable function having a single discontinuity point in the domain where it is defined, are utilized to derive analytical solutions for the radiation or scattering of oblique water waves by a fully extended porous barrier in both the cases of finite and infinite depths of water in two-layer fluid with constant densities. Also, complete analytical solutions are obtained for the boundary-value problems dealing with the generation or scattering of axi-symmetric water waves by a system of permeable and impermeable co-axial cylinders. Various results concerning the generation and reflection of the axisymmetric surface or interfacial waves are derived in terms of Bessel functions. The resonance conditions within the trapped region are obtained in various cases. Further, expansions for multipole-line-source oblique-wave potentials are derived for both the cases of finite and infinite depth depending on the existence of the source point in a two-layered fluid.

Dorsal Transscaphoid-Transtriquetral Perilunate Dislocation in Pseudarthrosis of the Scaphoid

A patient reported that he sustained a minor fall on the outstretched hand in hyperextension, pronation, and in ulnar deviation. Initial radiographs suggested dorsal transscaphoid-transtriquetral perilunate dislocation. Traditionally, however, this injury is the result of a high-energy impact. A CT scan obtained after closed reduction of the dislocation revealed not only a fresh fracture of the triquetrum but also two corticalized fragments of the scaphoid. A former major fall on this hand and a normal scaphoid of the other hand made pseudarthrosis more likely than scaphoid bipartition. Arthrography revealed intact lunotriquetral and scapholunate ligaments, precluding the possibility of preexisting ligamentous instability. Pseudarthrosis of the scaphoid with a loss of scaphoid function as a mechanical tie-rod of the carpus is most likely responsible for this complex injury. This is the first clinical study that shows that single scaphoid discontinuity without preexisting ligamentous carpal instability may lead to complex perilunar dislocation in minor trauma.

Attractors from one dimensional Lorenz-like maps

In this paper we study the properties of expanding maps with a single discontinuity on a closed interval and the resultant dynamics. For such a map, there exists a compact invariant subset which shares a lot of common properties with classical attractors such as the topological transitivity of the restricted map and the density of the periodic points. The invariant set, with more conditions on the boundary, can be shown to have an isolating neighborhood, hence is a chaotic attractor in the strong sense. Not all such maps derive trapping regions, yet by perturbation, those non-attractors can be made to have a trapping region.

Geometrically non-linear and consistently linearized embedded strong discontinuity models for 3D problems with an application to the dissection analysis of soft biological tissues

Three different finite element formulations with embedded strong discontinuities are derived on the basis of the enhanced assumed strain method. According to the work by Jirásek and Zimmermann [Int. J. Numer. Methods Engrg. 50 (2001) 1269] they are referred to as statically optimal symmetric (SOS), kinematically optimal symmetric (KOS) and statically and kinematically optimal non-symmetric (SKON) formulations. The effect of the discontinuities are characterized by additional degrees of freedom on the element level. Modifications to the standard KOS and SKON formulations are proposed in order to achieve consistency with the employed type of a three-field Hu–Washizu principle under mode-I condition. Under this condition the formulation satisfies the internal compatibility at the discontinuity, i.e. the relation between the stress in the bulk material and the traction across the discontinuity surface, which is not the case for the classical KOS formulation. We propose a suitable explicit expression for a transversely isotropic traction law in form of a displacement–energy function and assume that softening phenomena in the cohesive zone are modeled by a damage law, which depends on the maximum gap displacement of the deformation path. A linearization of all quantities, which are related to the non-linear problem, leads to new closed form expressions. In particular, we focus attention on the linearization of the cohesive traction vector. The associated element residua and stiffness matrices are provided. Standard static condensation of the internal degree of freedom leads to a generalized displacement model. A comparative study of the modified formulations, carried out by means of two numerical examples, show the performance of the individual approach. We employ constant-strain tetrahedral elements with a single discontinuity embedded. Among the known stress locking phenomena associated with the SOS formulation, we recognized that the (non-symmetric) SKON formulation was not able to provide meaningful results for the dissection process of an arterial layer in three-dimensions on distorted meshes. For both numerical examples the (symmetric) KOS formulation seems to be most suitable for representing the embedded discontinuities.

A cohesive segments method for the simulation of crack growth

A numerical method for crack growth is described in which the crack is not regarded as a single discontinuity that propagates continuously. Instead, the crack is represented by a set of overlapping cohesive segments. These cohesive segments are inserted into finite elements as discontinuities in the displacement field by exploiting the partition-of-unity property of shape functions. The cohesive segments can be incorporated at arbitrary locations and orientations and are not tied to any particular mesh direction. The evolution of decohesion of the segments is governed by a cohesive law. The independent specification of bulk and cohesive constitutive relations leads to a characteristic length being introduced into the formulation. The formulation permits both crack nucleation and discontinuous crack growth to be modelled. The implementation is outlined and some numerical examples are presented.

Sound propagation in a refracting atmosphere above an impedance discontinuity

de Jongs formulation of sound propagation above a ground with a single impedance change has been extended to include effects of a refracting atmosphere and atmospheric turbulence. The theory is compared with a numerical algorithm based on a hybrid Boundary Integral Equation/Fast Field Program developed for predicting the propagation of sound in a refracting atmosphere above an uneven, discontinuous terrain. By using the analogy of sound diffraction over curved surfaces to atmospheric refraction over flat ground surfaces, the effect of temperature and wind velocity gradients in the presence of flat ground surfaces can be studied. Measurements of the excess attenuation of sound from a point source over a mixed impedance curved surface are carried out in an anechoic chamber as well as outdoor measurements over a tarmac–grass discontinuity. These measurements are compared with predictions based on the extended de Jong theory and the hybrid BIE/FFP algorithm in the nonturbulent case. Results show that where there is a single discontinuity between acoustically hard and finite impedance surfaces both models are found to give satisfactory agreement with measured data except when the discontinuity is midway between the source and the detector.

Regular arrays of GaN nanorods

Regular arrays of GaN nanorods have been synthesized by reacting gallium and ammonia using Au as a catalyst on a MgO single crystal substrate treated by chemical etching. They were characterized by field emission scanning electron microscopy, energy dispersive X-ray spectroscopy (EDX), selected area electron diffraction (SAED) and X-ray diffraction (XRD). EDX, XRD SAED indicated that the nanorods were wurtzite single crystal GaN. We suggest that regular tactic square shape with steps on the single crystal MgO and discontinuity of the Au film may play an important role during the formation of regular arrays of GaN nanorods.

Influence of microscopic defects in type-II superconducting thin films on the magnetic flux penetration

The magnetic flux penetration into thin type-II superconducting films with circular defects is investigated. The artificial circular defects (diameter $=40 \ensuremath{\mu}\mathrm{m})$ in an ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ thin film (thickness $\ensuremath{\approx}300 \mathrm{nm})$ were prepared by pulse-laser irradiation. The flux penetration into the zero-field-cooled superconducting film was visualized by means of the magneto-optic method. A stepwise increase of the external magnetic field allowed a detailed investigation of the influence of local defects on the flux penetration. For a magnetic field parallel to a long sample (longitudinal geometry) with a long cylindrical defect a single parabolic discontinuity line appears. Also in the case of a thin superconducting film exposed to a transverse magnetic field (transverse geometry), a single parabolic discontinuity line has been supposed in the vicinity of a local defect. On the contrary, our investigations show that the flux and current distribution around a single defect in a superconducting thin film can be determined not by a single, but by two discontinuity parabolas. In thin superconducting films in transverse geometry screening currents in the Meissner region $(j<{j}_{c})$ are present in contrast to extended infinitely long samples in the longitudinal geometry. We explain our experimental results by the influence of these Meissner screening currents on the temporal formation of the shape of an approaching flux front.

Fracture void structure: implications for flow, transport and deformation

This review focuses on studies of flow, transport and deformation processes at a scale of a single discontinuity. The paper provides an evaluation of: (1) various methods suggested for geometrical characterization of void structure; and (2) theoretical and practical problems arising from significant differences between the actual geometry of fracture void structure and its parallel plate representation. The use of an equivalent aperture concept is shown to be seriously misleading in: (a) evaluation of flow regime, and hence selection of appropriate flow laws; (b) correlating tracer and hydraulic tests, and assessment of solute transport properties; and (c) relating hydraulic and mechanical apertures, and predicting influence of stress perturbation and deformability.

Period-adding bifurcations and chaos in a periodically stimulated excitable neural relaxation oscillator.

The response of an excitable neuron to trains of electrical spikes is relevant to the understanding of the neural code. In this paper, we study a neurobiologically motivated relaxation oscillator, with appropriately identified fast and slow coordinates, that admits an explicit mathematical analysis. An application of geometric singular perturbation theory shows the existence of an attracting invariant manifold, which is used to construct the Fenichel normal form for the system. This facilitates the calculation of the response of the system to pulsatile stimulation and allows the construction of a so-called extended isochronal map. The isochronal map is shown to have a single discontinuity and be of a type that can admit three types of response: mode-locked, quasiperiodic, and chaotic. The bifurcation structure of the system is seen to be extremely rich and supports period-adding bifurcations separated by windows of both chaos and periodicity. A bifurcation analysis of the isochronal map is presented in conjunction with a description of the various routes to chaos in this system.

A perturbation technique for the prediction of the displacement of a line-driven plate with discontinuities

A novel technique is presented for obtaining approximate analytic expressions for an inhomogeneous line-driven plate. The equation of motion for the inhomogeneous plate is transformed, and the transform of the total displacement is written as a sum of the solution for a homogeneous line-driven plate plus a term due to the inhomogeneity. The result is an integral equation for the transform of the inhomogeneous contribution. This expression may in general be solved numerically. However, by introducing a small parameter into the problem, it may be solved approximately using perturbation techniques. This series may not be convergent, but its convergence may be improved using Pade approximation. Results are presented for the case of a single mass discontinuity, and a distribution of mass discontinuities.

5-aminolevulinic acid, but not 5-aminolevulinic acid esters, is transported into adenocarcinoma cells by system BETA transporters.

5-aminolevulinic acid (5-ALA) and its ester derivatives are used in photodynamic therapy as precursors for the formation of photosensitizers. This study relates to the mechanisms by which 5-ALA is transported into cells. The transport of 5-ALA has been studied in a human adenocarcinoma cell line (WiDr) by means of [14C]-labeled 5-ALA. The rate of uptake was saturable following Michaelis-Menten kinetics (K(m) = 8-10 mM and Vmax = 18-20 nmol.(mg protein x h)-1), and Arrhenius plot of the temperature-dependent uptake of 5-ALA was characterized by a single discontinuity at 32 degrees C. The activation energy was 112 kJ.mol-1 in the temperature range 15 degrees-32 degrees C and 26 kJ.mol-1 above 32 degrees C. Transport of 5-ALA was Na+ and partly Cl(-)-dependent. Stoichiometric analysis revealed a Na+:5-ALA coupling ratio of 3:1. With the exception of valine, methionine and threonine, zwitterionic and basic amino acids inhibited the transport of 5-ALA. 5-ALA methyl ester was not an inhibitor of 5-ALA uptake. The transport was most efficiently inhibited, i.e. by 65-75%, by the beta-amino acids, beta-alanine and taurine and by gamma-aminobutyric acid (GABA). Accordingly, 5-ALA, but not 5-ALA methyl ester, was found to inhibit cellular uptake of [3H]-GABA and [14C]-beta-alanine. Protoporphyrin IX (PpIX) accumulation in the presence of 5-ALA (0.3 mM) was attenuated 85% in the presence of 10 mM beta-alanine, while PpIX formation in cells treated with 5-ALA methyl ester (0.3 mM) or 5-ALA hexyl ester (4 microM) was not significantly influenced by beta-alanine. Thus, 5-ALA, but not 5-ALA esters, is transported by beta-amino acid and GABA carriers in this cell line.

THE ASYMPTOTIC PERIODICITY OF LORENZ MAPS

A Lorenz map f : I → I is a one dimensional piecewise monotone map with a single discontinuity c. Let C(f)=∪n≥0f-n(c) be the collection of all the preimages of c. Authors prove that if C’(f) is countable then there exists M such that Card(Ω(x)) ≤ M for all x ∈ I. If C’(f) is uncountable then Ω(x) is uncountable for some x ∈ I. So f is asymptotically periodic if and only if C’(f) is countable.

The identification of the piecewise homogeneous thermal conductivity of conductors subjected to a heat flow test

The identification of the piecewise homogeneous thermal conductivity of conductors with unknown location of the discontinuities is investigated using additional boundary and⧹or interior measurements of the temperature in heat-flow experiments. A mathematical analysis is performed for the steady-state case, whilst for the unsteady case a numerical method based on the boundary element method, as a direct solution procedure, combined with an ordinary least-squares technique subject to bounds on the variables, is employed for obtaining an inverse numerical solution. The sensitivity coefficients for the unspecified boundary conditions and for interior temperatures are calculated and show the need for interior measurement information to be imposed. For a heat conductor presenting a single discontinuity, it was found that, when the number of time measurements is limited then two interior temperature measurements are necessary and sufficient in order to render a good estimate of the exact solution, otherwise one may increase the number of time measurements at a single interior sensor location to render the same result.

Thermal acclimation of phase behavior in plasma membrane lipids of rainbow trout hepatocytes.

The fluorescent probes laurdan (6-dodecanoyl-2-dimethylaminonapthalene) and N-[7-nitrobenz-2-oxa-1, 3-diazol-4-yl] dipalmitoyl-L-alpha-phosphatidylethanolamine (NBD-PE) in addition to Fourier transform infrared spectroscopy (FTIR) were employed to measure the phase behavior and physical properties of hepatocyte plasma membranes isolated from the livers of thermally acclimated (5 and 20 degreesC) rainbow trout (Oncorhynchus mykiss). The primary objective was to determine the extent to which the phase behavior of membrane lipids is conserved at different growth temperatures. Arrhenius plots of laurdan-generalized polarization revealed a single discontinuity believed to reflect either the onset of the gel-fluid phase transition or the formation of gel phase microdomains, and this discontinuity occurred at significantly higher temperatures in membranes of 20 degrees C (13.2 +/- 0.7 degrees C)- than 5 degrees C (7.2 +/- 0.1 degrees C)-acclimated trout. Similarly, acclimation from 5 to 20 degrees C increased both the onset temperature (from 2.0 +/- 0.3 to 7.2 +/- 0.6 degrees C) and the thermal range (from 10.9 +/- 0.5 to 16.0 +/- 1.0) of the gel-fluid transition as assessed by FTIR. The gel-fluid transition midpoint (approximately -2 degrees C) and completion temperatures (-9 degrees C) were unchanged by thermal acclimation. The anisotropy of NBD-PE fluorescence displayed a distinct minimum in membranes of both warm- and cold-acclimated trout (reflecting alterations in lipid packing that in pure lipid membranes ultimately lead to the formation of nonlamellar phases) in the range of 56-58 degrees C; only membranes of 5 degrees C-acclimated trout displayed an additional minimum at significantly lower temperatures (24.5 +/- 1.7 degrees C). Collectively, these data suggest that the regulation of both the temperature at which gel phase lipids begin to form in response to cooling as well as the propensity of membrane lipids to form nonlamellar phases at higher temperatures may be key features of membrane organization subject to adaptive regulation.

Similarity solution for capillary redistribution of two phases in a porous medium with a single discontinuity

In this paper we consider one-dimensional capillary redistribution of two immiscible and incompressible fluids in a porous medium with a single discontinuity. We study a special time-dependent solution, a similarity solution, which is found when the initial saturation is discontinuous at the same point as the permeability and porosity, and is constant elsewhere. The similarity solution can be used to validate numerical algorithms describing two-phase flow in porous media with discontinuous heterogeneities. We discuss the construction of the similarity solution, in which we pay special attention to the interface conditions at the discontinuity, both for media with positive and zero entry pressure. Moreover, we discuss some qualitative properties of the solution, and outline a numerical procedure to determine its graph. Examples are given for the Brooks-Corey and Van Genuchten model. We also consider similarity solutions for unsaturated water flow, which is a limit case of two-phase flow for negligible nonwetting phase viscosity.

Ergodicity for piecewise smooth cocycles over toral rotations

Let α be an ergodic rotation of the d-torus T = R/Z. For any piecewise smooth function f : T → R with sufficiently regular pieces the unitary operator V h(x) = exp(2πif(x))h(x + α) acting on L2(Td) is shown to have a continuous nonDirichlet spectrum if the gradient of f has nonzero integral. In particular, the resulting skew product Sf : Td+1 → Td+1 must be ergodic. If in addition α is sufficiently well approximated by rational vectors and f is represented by a linear function with noninteger coefficients then the spectrum of V is singular. In the case d = 1 our technique allows us to extend Pask’s result on ergodicity of cylinder flows on T× R to arbitrary piecewise absolutely continuous real-valued cocycles f satisfying T f = 0 and T f ′ 6= 0. Introduction. Let f be a piecewise absolutely continuous function on the unit interval such that T f ′ 6= 0 and let α ∈ R \Q. In [7] it is shown that the skew product transformation of T2 defined by Sf (x, y) = (x+ α, y + f(x)), where the addition is modulo 1, has a continuous spectrum on the orthocomplement in L2(T2) of the functions depending only on the first variable. Moreover, if f has a single discontinuity with jump c then for every k ∈ Z such that kc is not an integer the spectrum is singular on the invariant subspace Hk = {h(x) exp(2πiky) : h ∈ L2(T)} (see Theorem 2 of [7]; special cases have been proved earlier in [2] and [9]) . By investigating the invariant subspaces Hk the spectral analysis of the skew product reduces to that of the unitary operator V = Vα,f on L2(T), where (V h)(x) = e2πif(x)h(x+ α) (see Anzai [1]), and in fact to the spectral measure σα,f = σ of the function h = 1: 1991 Mathematics Subject Classification: Primary 28D05. Supported in part by KBN grant 2P 03A 07608.

An inverse problem to determine the piecewise homogeneous hydraulic conductivity within rocks

Abstract The experimental measurement of the piecewise homogeneous hydraulic conductivity of a rock sample in which the location of the discontinuities is unknown appears difficult, even in a one-dimensional situation. However, the values of these quantities can be obtained numerically using additional boundary and/or interior measurements of the pressure from transient hydraulic experiments. In the direct problem, the hydraulic conductivity is known, and only a solution for the pressure field is sought. However, in the inverse formulation of the problem, both the hydraulic conductivity and the pressure are unknown and have to be determined using additional pressure measurements. The numerical method employed for solving the diffusion equation is based on the boundary element method as a direct solution procedure, combined with an ordinary least-squares technique. The sensitivity coefficients are calculated for the unspecified boundary conditions and for interior pressures, and clearly show the need for interior measurement information to be imposed on the solution of the inverse ill-posed problem. The uniqueness of the solution of the inverse problem is thoroughly numerically investigated using additional pressure measurements at several prescribed times from various numbers of wells and different well locations. For a material presenting a single discontinuity in the hydraulic conductivity, and subject to certain experimental conditions, it was found that when the number of time measurements is limited then two interior well pressure measurements are necessary and sufficient in order to render a good estimate of the exact solution. Otherwise, it is necessary to increase the number of time measurements at a single interior well location to accomplish the same result.

Models and measurements of sound propagation from a point source over mixed impedance ground

Measurements of the excess attenuation of sound from a point source over mixed impedance ground in an anechoic chamber are compared with predictions obtained from models based on (a) the semiempirical theory due to De Jong, (b) Nyberg’s theory, (c) a Fresnel-zone approximation, and (d) a boundary element code. The impedance discontinuities studied in this work are perpendicular to the source–receiver line. When there is a single discontinuity between acoustically hard and finite impedance surfaces, the De Jong semiempirical model, the Fresnel-zone model and boundary element code are found to give satisfactory agreement with measured data. The frequency of the first maximum in attenuation is found to be highest at approximately 70% hard surface cover rather than at the 100% expected. It is argued that this is a result of edge diffraction. When extended to multiple impedance discontinuities, the De Jong semiempirical model performs poorly. However, both Nyberg’s theory and the boundary element code give good agreement with measured results, and the Fresnel-zone model gives qualitative agreement with the measured data. The measurements confirm that, under certain constraints given by Nyberg’s theory, the ground effect due to mixed impedance may be determined from that predicted by using the area-average impedance.

Geometric Shock-Capturing ENO Schemes for Subpixel Interpolation, Computation and Curve Evolution

Subpixel methods that locate curves and their singularities, and that accurately measure geometric quantities, such as orientation and curvature, are of significant importance in computer vision and graphics. Such methods often use local surface fits or structural models for a local neighborhood of the curve to obtain the interpolated curve. Whereas their performance is good in smooth regions of the curve, it is typically poor in the vicinity of singularities. Similarly, the computation of geometric quantities is often regularized to deal with noise present in discrete data. However, in the process, discontinuities are blurred over, leading to poor estimates at them and in their vicinity. In this paper we propose a geometric interpolation technique to overcome these limitations by locating curves and obtaining geometric estimates while (1) not blurring across discontinuities and (2) explicitly and accurately placing them. The essential idea is to avoid the propagation of information across singularities. This is accomplished by a one-sided smoothing technique, where information is propagated from the direction of the side with the “smoother” neighborhood. When both sides are nonsmooth, the two existing discontinuities are relieved by placing a single discontinuity, or shock. The placement of shocks is guided by geometric continuity constraints, resulting in subpixel interpolation with accurate geometric estimates. Since the technique was originally motivated by curve evolution applications, we demonstrate its usefulness in capturing not only smooth evolving curves, but also ones with orientation discontinuities. In particular, the technique is shown to be far better than traditional methods when multiple or entire curves are present in a very small neighborhood.