一类三阶拟线性方程组的奇摄动边值问题

本文用间接的方法证明了一类三阶拟线性方程组的奇摄动边值问题解的存在唯一性,并给出了解对小参数任意精确度的一致有效展开。     

Simulation of the distribution of electronic potential in two- dimensional anisotropic resistive-capacitive medium

The work was centered on theoretical investigation of the spread of electrotonic potential in the cardiac tissue during the delivery of long-term rectangular pre-threshold current impulses to the tissue. A mathematical model of two-dimensional anisotropic medium, with a circumference-shaped current source, has been developed. In accordance with the superposition principle well-known in electrostatics, the source of current was approximated by point sources placed on the perimeter of the current source in two ways: 1) by regular division of the perimeter of the current source in metric coordinate system, 2) by regular division of the perimeter of current source in the normalized coordinate system. Computerized programs have been developed for the computation of the distribution of electrotonic potential in the two-dimensional anisotropic medium. Families of curves were established evaluating the dependence of half-time (time during which electrotonic potential reaches half the stationary amplitude) on the size of the current source, anisotropy, the distance between the center of the current source and the point of registration (calculation) of electrotonic potential. Near the current source the dependence of half-time on the distance declines from linear.     

Analytical Approach to Predicting Temperature Fields in Multilayered Pavement Systems

An accurate and rapid estimation of the pavement temperature field is desired to better predict pavement responses and for pavement system design. In this paper, an innovative method to derive the theoretical solution of an axisymmetric temperature field in a multilayered pavement system is presented. The multilayered pavement system was modeled as a two-dimensional heat transfer problem. The temperature at any location (r,z) and any time t in an N-layer pavement system can be calculated by using the derived analytical solution. The Hankel integral transform with respect to the radial coordinate is utilized in the derivation of the solution. The interpolatory trigonometric polynomials based on discrete Fourier transform are used to fit the measured air temperatures and solar radiation intensities during a day, which are essential components in the boundary condition for the underlying heat transfer problem. A FORTRAN program was coded to implement this analytical solution. Measured field temperature results from a rigid pavement system demonstrate that the derived analytical solution generates reasonable temperature profiles in the concrete slab.     

Propagation Regimes of Fluid-Driven Fractures in Impermeable Rocks

This paper reviews recent results of a research program aimed at developing a theoretical framework to understand and predict the different modes of propagation of a fluid-driven fracture. The research effort involves constructing detailed solutions of the crack tip region, developing global models of hydraulic fractures for plane strain and radial geometry, and identifying the parameters controlling the fracture growth. The paper focuses on the propagation of hydraulic fractures in impermeable rocks. The controlling parameters are identified from scaling laws that recognize the existence of two dissipative processes: fracturing of the rock (toughness) and dissipation in the fracturing fluid (viscosity). It is shown that the two limit solutions (corresponding to zero toughness and zero viscosity) are characterized by a power law dependence on time and that the transition between these two asymptotic solutions depends on a single number, which can be chosen to be either a dimensionless toughness or a dimensionless viscosity. The viscosity- and toughness-dominated regime of crack propagation are then identified by comparing the general solutions with the asymptotic solutions. This analysis yields the ranges of the dimensionless parameter for which the solution can be approximated for all practical purposes either by the zero toughness or by the zero viscosity solution.     

Transverse Shear Stiffness of Composite Honeycomb Core with General Configuration

Based on the zeroth-order approximation of a two-scale asymptotic expansion, equivalent elastic shear coefficients of periodic structures can be evaluated via the solution of a local function τklij(y), and the homogenization process reduces to solving the local function τklij(y) by invoking local periodic boundary conditions. Then, effective transverse shear stiffness properties can be analytically predicted by reducing a local problem of a given unit cell into a 2D problem. In this paper, an analytical approach with a two-scale asymptotic homogenization technique is developed for evaluation of effective transverse shear stiffness of thin-walled honeycomb core structures with general configurations, and the governing 3D partial differential equations are solved with the assumptions of free warping constraints and constant variables through the core wall thickness. The explicit formulas for the effective transverse shear stiffness are presented for a general configuration of honeycomb core. A detailed study is given for three typical honeycomb cores consisting of sinusoidal, tubular, and hexagonal configurations, and their solutions are validated with existing equations and numerical analyses. The developed approach with certain modifications can be extended to other sandwich structures, and a summary of explicit solutions for the transverse shear stiffness of common honeycomb core configurations is provided. The lower bound solution provided in this study is a reliable approximation for engineering design and can be efficiently used for quick evaluation and optimization of general core configurations. The upper bound formula, based on the assumption of uniform shear deformation, is also given for comparison. Further, it is expected that with appropriate construction in the displacement field, the more accurate transverse stiffness can be analytically attained by taking into account the effect due to the face-sheet constraints.     

非线性界面动力学影响下颗粒增长的界面稳定性

研究了一个带有非线性温度依赖界面动力学的颗粒生长数学模型,分析颗粒界面的演化及其形态稳定性.利用渐近展开方法,获得颗粒生长的渐近解以及界面扰动的变化率.当界面动力学增加时,界面动力学过冷减小,颗粒增长速度也减小,非线性温度依赖界面动力学使得颗粒界面生长倾向于稳定.与忽略界面动力学的情形比较,非线性界面动力学显著减弱了球颗粒的生长速度.      

Methods of construction of an asymptotic theory for the optimum control over kinetic processes in technological objects

An illustrated conception of an asymptotic theory of optimum monitoring (OM) and a recurrent chain of OM systems that are modified by the Whitham method for the states perturbed by the coordinate and for singularly distributed perturbed OM systems is given. It is demonstrated that the recurrent OM chain obtained in this case can be rather simply resolved for each of its components separately.     

A parallel noise-robust algorithm to recover depth information from radial flow fields

A parallel algorithm operating on the units ('neurons') of an artificial retina is proposed to recover depth information in a visual scene from radial flow fields induced by ego motion along a given axis. The system consists of up to 600 radii with fewer than 65 radially arranged neurons on each radius. Neurons are connected only to their nearest neighbors, and they are excited as soon as a sufficiently strong gray-level change occurs. The time difference of two subsequently activated neurons is then used by the last-excited neuron to compute the depth information. All algorithmic calculations remain strictly local, and information is exchanged only between adjacent active neurons (except for the final read-out). This, in principle, permits parallel implementation. Furthermore, it is demonstrated that the calculation of the object coordinates requires only a single multiplication with a constant, which is dependent on only the retinal position of the active neuron. The initial restriction to local operations makes the algorithm very noise sensitive. In order to solve this problem, a predication mechanism is introduced. After an object coordinate has been determined, the active neuron computes the time when the next neuronal excitation should take place. This estimated time is transferred to the respective next neuron, which will wait for this excitation only within a certain time window. If the excitation fails to arrive within this window, the previously computed object coordinate is regarded as noisy and discarded. We will show that this predictive mechanism relies also on only a (second) single multiplication with another neuron-dependent constant. Thus, computational complexity remains low, and noisy depth coordinates are efficiently eliminated. Thus, the algorithm is very fast and operates in real time on 128 x 128 images even in a serial implementation on a relatively slow computer. The algorithm is tested on scenes of growing complexity, and a detailed error analysis is provided showing that the depth error remains very low in most cases. A comparison to standard flow-field analysis shows that our algorithm outperforms the older method by far. The analysis of the algorithm also shows that it is generally applicable despite its restrictions, because it is fast and accurate enough such that a complete depth percept can be composed from radial flow field segments. Finally, we suggest how to generalize the algorithm, waiving the restriction of radial flow.     

Stochastic epidemics: the probability of extinction of an infectious disease at the end of a major outbreak

The aim of this study is to derive an asymptotic expression for the probability that an infectious disease will disappear from a population at the end of a major outbreak ('fade-out'). The study deals with a stochastic SIR-model. Local asymptotic expansions are constructed for the deterministic trajectories of the corresponding deterministic system, in particular for the deterministic trajectory starting in the saddle point. The analytical expression for the probability of extinction is derived by asymptotically solving a boundary value problem based on the Fokker-Planck equation for the stochastic system. The asymptotic results are compared with results obtained by random walk simulations.     

Semianalytical Solution of Wave-Controlled Impact on Composite Laminates

Based on a structural model for wave-controlled impact, a modified Hertzian contact law was used to investigate the impact responses of composite laminates. The original nonlinear governing equation was transformed into two linear equations using asymptotic expansion. Closed-form solution can be derived for the first linear homogeneous equation, which is the equation of motion for single degree of freedom system with viscous damping. The second linear nonhomogeneous equation was solved numerically. The overall impact responses for wave-controlled impacts can be obtained semianalytically and agree well with the numerical solutions of nonlinear governing equations. The proposed methodology is useful for providing guidance to numerical simulation of impact on complex composite structures with contact laws fitting from experimental data.     

Assessing the role of the biomaterial Aquavene in patient reactions to Landmark midline catheters

Generators of early cortical somatosensory evoked potentials (SEPs) still remain to be precisely localised. This gap in knowledge has often resulted in unclear and contrasting SEPs localisation in patients with focal hemispheric lesions. We recorded SEPs to median nerve stimulation in a patient with right frontal astrocytoma, using a 19-channel recording technique. After stimulation of the left median nerve, N20 amplitude was normal when recorded by the parietal electrode contralateral to the stimulation, while it was abnormally enhanced in traces obtained by the contralateral central electrode. The amplitude of the frontal P20 response was within normal limits. This finding suggests that two dipolar sources, tangential and radial to the scalp surface, respectively, contribute concomitantly to N20 generation. The possible location of the N20 radial source in area 3a is discussed. The P22 potential was also recorded with increased amplitude by the central electrode contralateral to the stimulation, while N30 amplitude was normal in frontal and central traces. We propose that the radial dipolar source of P22 response is independent from both N20 and N30 generators and can be located either in 3a or in area 4. This report illustrates the usefulness of multichannel recordings in diagnosing dysfunction of the sensorimotor cortex in focal cortical lesions.     

Investigation of the Damage-Dependent Response of Mooring Ropes

The focus of this paper is on the development of an analytical damage model for predicting the deterioration of the mechanical properties of polyester (PET) ropes subjected to static tension loading. Experimental data on small PET ropes are used to estimate the evolution of damage using the effective stress concept and the principle of strain equivalence. The proposed damage model relies on an empirically based cumulative scalar damage function, which is founded on the assumption that the strain range experienced by rope elements is the main source of damage under static tension loading conditions. In this particular study, the evolution of the damage function is represented by both power law and polynomial forms. Based on experimental observations, softening behavior is developed by rope elements after reaching their maximum load-carrying capacities. This softening behavior is captured by the damage function through an asymptotic expansion technique (perturbation method). Comparisons between predicted rope responses and experimental data are provided to illustrate the use of the proposed damage model to estimate PET rope response.     

New Approximations for Reliability Integrals

A new asymptotic expansion is applied to approximate reliability integrals. The asymptotic approximation reduces the problem of evaluating a multidimensional probability integral to solving an unconstrained minimization problem. Approximations are developed in both the transformed (independently, normally distributed) variables and the original variables. In the transformed variables, the asymptotic approximation yields a very simple formula for approximating the value of the second-order reliability method integrals. In many cases, it may be computationally expensive to transform to normal variables, and an approximation using the probability distribution for the original variables can be used. Examples are presented illustrating the accuracy of the approximations, and results are compared with some existing approximations of reliability integrals.     

Free Vibrations of Taut Cable with Attached Damper. II: Nonlinear Damper

Free vibrations of a taut cable with a nonlinear power-law damper attached near the end are considered. An approximate analytical solution for the amplitude-dependent effective damping ratios in each mode is developed by assuming the same form of solution as for the linear damper and minimizing the mean-square error in the force equilibrium at the damper. An asymptotic approximate solution for small frequency shifts reveals a nondimensional grouping of parameters allowing the development of an amplitude-dependent “universal estimation curve” for the power-law damper. The shape of the universal curve is slightly different for each value of the damper exponent, but for a given exponent the curve is nearly invariant over the same range of parameters as the universal curve for the linear damper. This formulation yields insights into the dependence of nonlinear damper performance on mode number and amplitude of oscillation, suggesting potential advantages that may be offered by a nonlinear damper over a traditional linear damper.     

Z/I and A-band lattice spacings in frog skeletal muscle: effects of contraction and osmolarity

A-band and Z-line/I-band lattice spacings were measured by small-angle X-ray diffraction from relaxed and isometrically-contracting whole frog sartorius muscles with lattice spacings reduced or swollen by changing the osmolarity of the bathing solution. A-band spacing increased by approximately 3% upon isometric contraction at reduced lattice spacings (245-356 mOsm) and decreased by approximately 1% at swollen spacings (172 mOsm), similarly to the behaviour of skinned muscles upon changing from the relaxed state to rigor. The Z/I lattice underwent a significant lattice expansion (3-8%) upon isometric contraction at all osmolarities, in qualitative agreement (but quantitative disagreement) with results from electron microscopy on mammalian skeletal muscle. Lattice areas calculated for the Z/I and A-band lattices indicate a barrel-shaped sarcomere in the resting state, which may provide a partial explanation for how longitudinal forces produced in the A-band can produce a radial expansive force in the Z-line during contraction. The radial component of cross-bridge stiffness was calculated from the A-band data for contracting muscle, using a lattice stability model incorporating structural, osmotic and electrostatic forces. The calculations gave a radial cross-bridge stiffness during contraction of about 9 x 10(5) N m-2, and outward radial force per thick filament in normal Ringer's solution of 6 x 10(-9) N, corresponding to a radial force per cross-bridge of 10(-11) N.     

Boundary-Layer Effect in Composite Beams with Interlayer Slip

An apparent analytical peculiarity or paradox in the bending behavior of elastic-composite beams with interlayer slip, sandwich beams, or other similar problems subjected to boundary moments exists. For a fully composite beam subjected to such end moments, the partial composite model will render a nonvanishing uniform value for the normal force in the individual subelement. This is from a formal mathematical point of view in apparent contradiction with the boundary conditions, in which the normal force in the individual subelement usually is assumed to vanish at the extremity of the beam. This mathematical paradox can be explained with the concept of boundary layer. The bending of the partially composite beam expressed in dimensionless form depends only on one structural parameter related to the stiffness of the connection between the two subelements. An asymptotic method is used to characterize the normal force and the bending moment in the individual subelement to this dimensionless connection parameter. The outer expansion that is valid away from the boundary and the inner expansion valid within the layer adjacent to the boundary (beam extremity) are analytically given. The inner and outer expansions are matched by using Prandtl’s matching condition over a region located at the edge of the boundary layer. The thickness of the boundary layer is the inverse of the dimensionless connection parameter. Finite-element results confirm the analytical results and the sensitivity of the bending solution to the mesh density, especially in the edge zone with stress gradient. Finally, composite beams with interlayer slip can be treated in the same manner as nonlocal elastic beams. The fundamental differential equation appearing in the constitutive law associated with the partial-composite action in a nonlocal elasticity framework is discussed. Such an integral formulation of the constitutive equation encompassing the behavior of the whole of the beam allows the investigation of the mechanical problem with the boundary-element method.     

Prediction of liquid metal viscosities using an adjustable hard sphere radial distribution curve

Predictions of liquid metal viscosities have been made using the Born-Green equation. The pair potential was calculated by the Percus-Yevick and hypernetted chain equations with the predicted structure factor and was used with the Mie-Gruneisen potential to make the calculations. Instead of using an experimentally determined radial distribution curve, however, an adjustable hard sphere radial distribution curve is employed to obtain the pair potential. The first peak of the hard sphere radial distribution curve used here was adjusted to represent the radial distribution curve at the melting point for any liquid metal. This was done by using the number of nearest neighbors and the distance to the first maximum in the radial distribution curve. This is possible because of the similarities in the radial distribution curves of liquid metals. A temperature correction is employed so that the viscosities can be calculated over a wide temperature range. The results are compared with literature values.     

Thermal expansion and crown evaluations of rolls in rolling processes

An understanding of the thermal expansion of work rolls is essential to the proper control of the flatness of a rolled strip in operating a modern, high-speed rolling mill. In this paper, an analytical solution has been developed to provide the thermal displacements and expansions of the roll. The thermal expansion data, in turn, can be utilized to estimate the thermal crown for shape control and contributes to alleviating many shape problems of the rolled product. The thermal displacement potential function has been used to solve the governing partial differential equations for the thermal displacements. The general stress function is then obtained to modify the displacement potential solution to satisfy the boundary conditions of the free roll surface. To demonstrate the accuracy and reliability of the solution developed, the results of the present solution have been compared with previously published numerical and experimental results at typical rolling conditions; good agreement has been found between them. A parametric study is then performed to provide the required information for proper control of the roll and strip shape in operating a high-speed rolling mill.     

Two-dimensional cancellation neglect a review and suggested method of analysis

Directional bias on cancellation has thus far not been standardized. While cancellation tasks are primarily used to assess lateral performance asymmetries, they may also reveal two-dimensional (i.e., combined lateral and radial) neglect patterns. We propose a method to evaluate and report cancellation neglect regardless of whether the neglect pattern is strictly unilateral or two-dimensional. Our method establishes the location of the geographic center of all neglected stimuli relative to the page center by averaging their Cartesian coordinates. This "neglect center" is reported in polar coordinates to indicate its distance and direction from the page center. We apply our method to published examples of two-dimensional neglect. We find that neglect centers from different cancellation performances may not be statistically distinct even though they may occupy different quadrants. In addition, the net direction of neglect found by the coordinate method may differ from that inferred from measuring differences in quadrant omission totals. The suitability of the coordinate vs. the quadrant method will depend on the mechanism hypothesized for visuospatial exploration under particular test conditions. Using both approaches may detect different attentional biases operating during the same task. The coordinate method is appropriate for conventional cancellation testing. By incorporating the precise locations of all neglected stimuli and determining the net neglect direction in two dimensions, the technique may stimulate more comprehensive explanations for directional bias.