Determining the Profile of a Nonmonotonic Hill

In a well-known generalization of the tautochrone problem, the shape of a monotone hill is determined by observing the travel time of a particle on the surface of the hill. In this paper we present a technique for uniquely determining the shape of any smooth hill, whether monotone or not, by observing collisions of particles on the hill. It is shown that this inverse problem is well posed. Error bounds for the solution are given for the case in which there are incomplete data. The efficiency of these bounds is shown by means of an example.     

Effect of a dispersed filler on the modulus of elasticity of composites in the glassy state

An expression is proposed for predicting the reinforcement of composites by a dispersed filler in both the rubbery and glassy states with allowance for the internal thermal stresses in the polymer matrix. It is shown that by varying the plasticizer concentration in composites based on polyvinyl chloride it is possible to regulate the thermal stresses in the polymer matrix. The time-stress and time-concentration superposition principles are shown to be applicable to the dependence of the relative modulus of elasticity on filler concentration for glassy composites.     

Efficient estimation of the error distribution in a varying coefficient regression model

It is shown that the weighted residual-based estimator of Schick, Zhu, and Du (2017) is efficient in some special cases and can be made to be efficient by adding a stochastic correction term. The efficiency is shown by deriving the efficient influence function and establishing a uniform stochastic expansion with this influence function. The correction term relies on estimators of the score function for the errors and other characteristics of the model.     

Enforced stability of a simple eigenvalue in the continuous spectrum of a waveguide

It is shown in the paper that, under several orthogonality and normalization conditions and a proper choice of accessory parameters, a simple eigenvalue lying between thresholds of the continuous spectrum of the Dirichlet problem in a domain with a cylindrical outlet to infinity is not taken out from the spectrum by a small compact perturbation of the Helmholtz operator. The result is obtained by means of an asymptotic analysis of the augmented scattering matrix.     

Analysis of the generalization ability of a full decision tree

Classification algorithms based on full decision trees are investigated. Due to the decision tree construction under consideration, all the features satisfying a branching criterion are taken into account at each special vertex. The generalization ability of a full decision tree is estimated by applying margin theory. It is shown on real-life problems that the construction of a full decision tree leads to an increase in the margins of the learning objects; moreover, the number of objects with a positive margin increases as well. It is shown that the empirical Rademacher complexity of a full decision tree is lower than that of a classical decision tree.     

Attractors of the sine-Gordon equation in the field of a quasiperiodic external force

The well-known sine-Gordon equation, supplemented with small damping and small quasiperiodic external force, is studied under the zero Dirichlet boundary conditions at the endpoints of a finite interval. The main assumption is that all frequencies of the external force are in 1:1 resonance with certain eigenfrequencies of the unperturbed equation; i.e., the socalled fundamental multifrequency resonance is observed. It is shown that in this case, by an appropriate choice of the parameters of the external force, one can make it so that the boundary value problem has a stable invariant torus of any finite dimension that bifurcates from zero on any preassigned finite set of spatial modes. It is also shown (by numerical analysis) that in a number of cases the above-mentioned torus coexists with a chaotic attractor.     

On the regulation number of a multigraph

The regulation number of a multigraphG having maximum degreed is the minimum number of additional vertices that are required to construct ad-regular supermultigraph ofG. It is shown that the regulation number of any multigraph is at most 3. The regulation number of a multidigraph is defined analogously and is shown never to exceed 2. A multigraphG has strengthm if every two distinct vertices ofG are joined by at mostm parallel edges. For a multigraphG of strengthm and maximum degreed, them-regulation number ofG is the minimum number of additional vertices that are required to construct ad-regular supermultigraph ofG having strengthm. A sharp upper bound on the 2-regulation number of a multigraph is shown to be (d+5)/2, and a conjecture for generalm is presented. Research supported by a Western Michigan University faculty research fellowship. Research Professor of Electrical Engineering and Computer Science, Stevens Institute, Hoboken, NJ and Visiting Scholar, Courant Institute, New York University, Spring 1984. Research supported in part by a Western Michigan University research assistantship from the Graduate College and the College of Arts and Sciences.     

Optimization of the motion in a resistant medium of a body with a movable internal mass

The motion, in a resistant medium, of a system consisting of a rigid body and movable internal mass is considered. The external medium acts on the body by a force that piecewise linearly depends on its speed. The class of periodic motions of the internal mass for which the speed of this mass relative to the body is piecewise constant is studied. It is shown that, under certain conditions, the forward movement of the whole system in the medium is possible. The average speed of this movement over a period is determined. Optimal parameters of the motion of the internal mass for which the average speed of the system movement is maximal are found.     

On the complexity of a cyclic shift of a set of real numbers

The complexity of implementing a cyclic shift of a 2ⁿ-tuple of real numbers by Boolean circuits over the basis consisting of a ternary choice function and all binary Boolean functions is shown to be 2ⁿ n.     

Existence of solutions and stability of a class of parabolic systems perturbed by generalized white noise on the boundary ∗

This paper is concerned with a class of stochastic boundary value problems and their stability questions. The system, we consider, is governed by a parabolic partial differential equation perturbed by generalized white noise on the boundary. Existence of weak solutions and their regularity properties are established. It is also shown that the solution of the autonomous system generates a Feller process in a Hilbert space, in case the spatial operator is time invariant. The questions of Lyapunov type stability of this class of systems are also examined. The system is shown to be almost surely globally asymptotically stable with respect to a ball centered at the origin. Further, it is shown that there exists a measure, supported on the attractor, which is invariant with respect to the adjoint Feller semigroup. An explicit expression for the generator of the semigroup is also given     

Variance of a standard vector Monte Carlo estimate in the theory of polarized radiative transfer

The spectral radius ρ of the matrix integral operator defining the covariance matrix of a standard vector Monte Carlo estimate in the polarized radiative transfer problem is examined. The theory of positive operators is used to analytically calculate ρ = ρ₀ for transfer through an infinite homogeneous medium. For a bounded medium, it is shown that ρ is approximately equal to ρ₀ times the spectral radius of the operator corresponding to radiative transfer without polarization. This is shown numerically by estimating the iterations of the corresponding resolvent and approximately analytically by using a perturbation of a special functional.     

On the sensitivity of a Euclidean projection

The structure and behavior of Euclidean projections of a point onto a set defined by parametric constraints is studied. Under the Mangasarian-Fromovitz constraint qualification, it is shown that the projection is locally unique and continuous and, if the feasible set is constant, locally Lipschitz continuous as well. Quantitative results are obtained characterizing the asymptotic behavior of projections under perturbations in a given direction.     

The stability of the steady motion of a gyrostat with a liquid in a cavity

A symmetrical rigid body with a spherical base, carrying a rotor and having a cavity in the shape of an ellipsoid of revolution, completely filled with an ideal incompressible liquid in uniform vortex motion, is moving along an absolutely rough plane. It is shown that this system admits of an energy integral, Jellett's integral, the integral of constant vorticity and a geometric integral. The construction of a Lyapunov function as a linear combination of first integrals [1] yields the sufficient conditions for the rotation of the gyrostat about the vertically positioned axis of symmetry to be stable. The conditions for the gyrostat's rotation to be unstable are found. It is shown that the rotor may prove to have either a stabilizing or destabilizing effect on the system and that the gyrostat admits of motions of the type of regular precession. The sufficient conditions for the stability of these motions are obtained.     

Stability analysis of a difference scheme for the Navier-Stokes equations

An application of stability analysis by the energy method is made to a practical problem of unsteady viscous flow solved numerically by Fromm. The practical stability criterion for the scheme is determined and a rigorous proof of convergence to a smooth solution is given. It is also shown how to construct an energy for any 3-level scalar difference equation.     

On the Boundary Condition at the Surface of a Porous Medium

A theoretical justification is given for an empirical boundary condition proposed by Beavers and Joseph [1]. The method consists of first using a statistical approach to extend Darcy's law to non homogeneous porous medium. The limiting case of a step function distribution of permeability and porosity is then examined by boundary layer techniques, and shown to give the required boundary condition. In an Appendix, the statistical approach is checked by using it to derive Einstein's law for the viscosity of dilute suspensions.     

The scattering of a plane wave by a trap in the critical case

The scattering of a plane wave by a resonator with a narrow coupling channel is considered. The velocity potential of the scattered wave in this resonator has two series of poles with small imaginary parts, corresponding to the main trap and the coupling channel, the effect of which inside the trap differs by an order of magnitude. The critical case, when the limiting value for the poles from both series is the same, is investigated. It is shown that in this case two poles exist, which converge to this limiting value, and they both inherit resonance properties, characteristic for poles generated by the main trap. The principal terms of the asymptotic forms of the poles and the scattered wave are constructed.     

Investigation of a mixing length and a two-equation turbulence model utilizing the finite element method

The finite element method is used to analyse turbulent coaxial jet flow using the mixing length viscosity model. The predictions from the model are shown to compare favourably with the results of experiment and of a finite difference based method. The implementation of a two-equation turbulence model in the finite element context is also discussed and the feasibility of such an approach is demonstrated by an analysis of fully developed turbulent flow in smooth walled pipes and channels.     

On the motion of a heavy body with a circular base on a horizontal plane and riddles of curling

In this paper we discuss the dynamics of an axisymmetric rigid body whose circular area moves upon a horizontal rough surface. We investigate the interaction between the character of the law of friction and the curvature of the body’s trajectory. For the case of a curling stone it is shown that the observed effects can only be explained using the dependence of the friction coefficient on the Gümbel number. The procedure for constructing the law of friction based on experimental data is developed. It is shown that the available data can only be substantiated by means of anisotropic friction. The simplest model of such friction is constructed which provides quantitative coincidence with the experiment.     

On the hydrodynamic limit of a Ginzburg-Landau lattice model

Summary A lattice system of interacting diffusion processes is investigated. The evolution is attractive and time reversible, the spin satisfies a conservation law. It is shown that the rescaled spin field converges in probability to the corresponding solution to a nonlinear diffusion equation.Supported in part by the Hungarian National Foundation for Scientific Research, grant No. 819/1, and by the Mathematical Department of Rutgers University, N.S.F. grant DMR 8612369     

Spreading of the free boundary of an ideal fluid in a vacuum

The diameter of a region occupied by an ideal fluid surrounded by vacuum will be shown to grow linearly in time provided the pressure is positive and there are no singularities. A family of explicit spherically symmetric, self-similar global solutions is constructed which illustrate the result in the compressible case.