Statistical Inference of Left Truncated and Right Censored Data from Marshall–Olkin Bivariate Rayleigh Distribution

In this paper, statistical inference and prediction issue of left truncated and right censored dependent competing risk data are studied. When the latent lifetime is distributed by Marshall–Olkin bivariate Rayleigh distribution, the maximum likelihood estimates of unknown parameters are established, and corresponding approximate confidence intervals are also constructed by using a Fisher information matrix and asymptotic approximate theory. Furthermore, Bayesian estimates and associated high posterior density credible intervals of unknown parameters are provided based on general flexible priors. In addition, when there is an order restriction between unknown parameters, the point and interval estimates based on classical and Bayesian frameworks are discussed too. Besides, the prediction issue of a censored sample is addressed based on both likelihood and Bayesian methods. Finally, extensive simulation studies are conducted to investigate the performance of the proposed methods, and two real-life examples are presented for illustration purposes.

Mechanics of side-slipping in alpine skiing. Braking and skidded traversing

A recently proposed simple approximate theory of snow machining is applied to modelling of several basic manoeuvres of alpine skiing: fall-line side-slipping, traversing, and hockey stop. The results agree with the skiing practice and explain the abnormally high friction reported in previous field studies. They also prepare foundation for future rigorous testing of the theory, which will determine its accuracy and limits of applicability.

Effects of Shear Lag in Steel Box Girders of a Crane Runway

The term of shear lag is related to the discrepancies between the approximate theory of the bending of beams and their real behaviour. It refers to the increases of the bending stresses near the flange-to-web junctions and the corresponding decreases in the flange stresses away from these junctions. In the case of wide flanges of plated structures, shear lag caused by shear strains, which are neglected in the conventional theory, may be taken into account by a reduced flange width concentrated along the webs of the steel girders. In EN 1993-1-5, the concept of taking shear lag into account is based on effective width of the flange which is defined in order to have the same total normal force in the gross flange subjected to the real transverse stress distribution as the effective flange subjected to a uniform stress equal to the maximum stress of the real transverse distribution. Some aspects concerning the shear lag phenomenon and a working example for a box girder of a heavy crane runway to illustrate the determination of the shear lag effect are also presented.

Multiscale Method for Solving High-order BVPs

This paper presents a numerical algorithm for solving high-order BVPs. We introduce the construction method of multiscale orthonormal basis in Wm[0; 1] by multiscale orthonormal basis in W1[0; 1]. We define approximate solution, and obtain the approximate solution of high-order BVPs by using the approximate theory. Moreover, the convergence and stability of the algorithm are improved. At last, several numerical experiments show the feasibility of the proposed method.

Snow machining and side slipping in alpine skiing

A simple approximate theory of snow machining is developed and tested against the results of past laboratory experiments. It is also applied to side-slipping and traversing in alpine skiing, and yields realistic predic- tions which can be tested experimentally on real ski slopes. Eventually, the theory could be used in modelling of skiing turns involving the phaseof skidding.

Mechanics of wedge turns in alpine skiing

A simple approximate theory of snow machining is applied to modelling successive wedge turns of alpine skiing. The results are in agreement with available experimental investigations of such turns. In particular, the model explains the abnormally high values for the coefficient of friction reported in these studies.

Extraction of density-layered fluid from a porous medium

We consider axisymmetric flow towards a point sink from a stratified fluid in a vertically confined aquifer. We present two approaches to solve the equations of flow for the linear density gradient case. Firstly, a series method results in an eigenfunction expansion in Whittaker functions. The second method is a simple finite difference method. Comparison of the two methods verifies the finite difference method is accurate, so that more complicated nonlinear, density stratification can be considered. Such nonlinear profiles cannot be considered with the eigenfunction approach. Interesting results for the case where the density stratification changes from linear to almost two-layer are presented, showing that in the nonlinear case there are certain values of flow rate for which a steady solution does not occur. References Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions, 9th ed. National Bureau of Standards, Washington, 1972. Bear, J. and Dagan, G. Some exact solutions of interface problems by means of the hodograph method. J. Geophys. Res. 69(8):1563–1572, 1964. doi:10.1029/JZ069i008p01563 Bear, J. Dynamics of fluids in porous media. Elsevier, New York, 1972. https://store.doverpublications.com/0486656756.html COMSOL Multiphysics. COMSOL Multiphysics Programming Reference Manual, version 5.3. https://doc.comsol.com/5.3/doc/com.comsol.help.comsol/COMSOL_ProgrammingReferenceManual.pdf Farrow, D. E. and Hocking, G. C. A numerical model for withdrawal from a two layer fluid. J. Fluid Mech. 549:141–157, 2006. doi:10.1017/S0022112005007561 Henderson, N., Flores, E., Sampaio, M., Freitas, L. and Platt, G. M. Supercritical fluid flow in porous media: modelling and simulation. Chem. Eng. Sci. 60:1797–1808, 2005. doi:10.1016/j.ces.2004.11.012 Lucas, S. K., Blake, J. R. and Kucera, A. A boundary-integral method applied to water coning in oil reservoirs. ANZIAM J. 32(3):261–283, 1991. doi:10.1017/S0334270000006858 Meyer, H. I. and Garder, A. O. Mechanics of two immiscible fluids in porous media. J. Appl. Phys., 25:1400–1406, 1954. doi:10.1063/1.1721576 Muskat, M. and Wycokoff, R. D. An approximate theory of water coning in oil production. Trans. AIME 114:144–163, 1935. doi:10.2118/935144-G GNU Octave. https://www.gnu.org/software/octave/doc/v4.2.1/ Yih, C. S. On steady stratified flows in porous media. Quart. J. Appl. Maths. 40(2):219–230, 1982. doi:10.1090/qam/666676 Yu, D., Jackson, K. and Harmon, T. C. Disperson and diffusion in porous media under supercritical conditions. Chem. Eng. Sci. 54:357–367, 1999. doi:10.1016/S0009-2509(98)00271-1 Zhang, H. and Hocking, G. C. Axisymmetric flow in an oil reservoir of finite depth caused by a point sink above an oil-water interface. J. Eng. Math. 32:365–376, 1997. doi:10.1023/A:1004227232732 Zhang, H., Hocking, G. C. and Seymour, B. Critical and supercritical withdrawal from a two-layer fluid through a line sink in a bounded aquifer. Adv. Water Res. 32:1703–1710, 2009. doi:10.1016/j.advwatres.2009.09.002 Zill, D. G. and Wright, W. S. Differential Equations with Boundary-value problems, 8th Edition. Brooks Cole, Boston USA, 2013.

Transformation of envelope solitons propagating over a bottom step

<p>The problem of the weakly nonlinear wave transformation on a bottom step is studied analytically and numerically by means of the direct simulation of the Euler equation. It is assumed that the quasi-linear wave packets can be described by the nonlinear Schrödinger equation for surface waves in finite-depth water. The process of wave transformation in the vicinity of the bottom step can be described within the framework of the linear theory and the transformation coefficients (the transmission and reflection coefficients) can be determined by the approximate formula suggested in [1]. The fate of transmitted and reflected wave trains emerging from the incident envelope soliton can be determined with the help of the Inverse Scattering Technique [2, 3].</p><p>The parameters of secondary envelope solitons (their number, amplitudes, and speeds) asymptotically forming in the far-field zone are obtained analytically and compared against the numerically calculated ones, as the functions of the depth drop <em>h</em><sub>2</sub>/<em>h</em><sub>1</sub>, where <em>h</em><sub>1</sub> and <em>h</em><sub>2</sub> are the undisturbed water depths in front of and behind the bottom step, respectively. It is shown that the wave amplitudes can notably increase when the envelope soliton travels from the relatively shallow to much deeper water. The amplitudes of secondary solitons can exceed more than twice the amplitude of the incident wave.</p><p>The direct numerical simulation of envelope soliton transformation was undertaken by means of the High Order Spectral Method [4, 5]. The comparison of approximate analytical solutions with the results of numerical simulations reveals the domains of very good agreement between the data where the approximate theory is applicable. In the meantime, the noticeable disagreement between the approximate nonlinear theory and the direct simulations is found when the theory is inapplicable.</p><p>The research by A.S. is supported by the RFBR grant No. 18-02-00042; he also acknowledges the support from the International Visitor Program of the University of Sydney and is grateful for the hospitality of the University of Southern Queensland. The research of Y.S. was support by the grant of the President of the Russian Federation for State support of scientific research of leading scientific Schools of the Russian Federation NSh-2485.2020.5.</p><p>[1] Kurkin, A.A., Semin, S.V., and Stepanyants, Yu.A., Transformation of Surface Waves over a Bottom Step. Izvestiya, Atmospheric and Oceanic Physics, 2015, Vol. 51, 214–223.</p><p>[2] Zakharov, V.E., Shabat, A.B., Exact theory of two-dimensional self-focussing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP, 1972, Vol. 34, 62-69.</p><p>[3] Slunyaev, A., Klein, M., Clauss, G.F., Laboratory and numerical study of intense envelope solitons of water waves: generation, reflection from a wall and collisions. Physics of Fluids, 2017, Vol. 29, 047103.</p><p>[4] West, B.J., Brueckner, K.A., Janda, R.S., Milder, D.M., Milton, R.L., A new numerical method for surface hydrodynamics. J. Geophys. Res., 1987, Vol. 92, 11803-11824.</p><p>[5] Ducrozet, G., Gouin, M., Influence of varying bathymetry in rogue wave occurrence within unidirectional and directional sea-states. J. Ocean Eng. Mar. Energy, 2017, Vol. 3, 309-324.</p>

Investigation of Rayleigh wave interaction with surface defects

The current article is concerned with the interaction of Rayleigh waves with surface defects of arbitrary shape in a homogeneous, isotropic, linearly elastic half-space. Using a linear superposition principle, the interaction generates a scattered field which is equivalent to the field radiated from a distribution of horizontal and vertical tractions on the surface of the defect. These tractions are equal in magnitude but opposite in sign to the corresponding tractions obtained from the incident wave. The scattered field is then computed as the superposition of the displacements radiated from the tractions at every point of the defect surface using the reciprocity theorem approach. The far-field vertical displacements are compared with calculations obtained by the boundary element method (BEM) for circular, rectangular, triangular and arbitrary-shaped defects. Comparisons between the theoretical and BEM results, which are graphically displayed, are in excellent agreement. It is also discussed the limitations of the proposed approximate theory. Keywords: half-space; Rayleigh wave; surface defect; reciprocity theorem; boundary element method (BEM).