Intersection area of a stationary circle and a circle under a coupled axial rotation and translation

A circle is placed concentrically in a circle of equal or larger size. The circle is then rotated along a vertical axis, creating an ellipse, and translated along the horizontal axis. The intersection area of the circle and circle/ellipse is determined as function of the rotation angle and the relative size of the initial circles. This configuration corresponds to the closing of a ball valve used to control the flow of fluids through pipes.     

Two‐weight norm inequalities for multilinear potential type integral operators

We give a condition which is sufficient for the two‐weight (p, q) inequalities for multilinear potential type integral operators, where 1 < p ≤ q < ∞. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Limit velocity for a driven particle in a random medium with mass aggregation

We study a one-dimensional infinite system of particles driven by a constant positive force F which acts only on the leftmost particle which is regarded as the tracer particle (t.p.). All other particles are field neutral, do not interact among themselves, and independently of each other with probability 0<p≤1 are either perfectly inelastic and “stick” to the t.p. after the first collision, or with probability 1−p are perfectly elastic, mechanically identical and have the same mass m. At initial time all particles are at rest, and the initial measure is such that the interparticle distances ξ_i's are i.i.d. r.v.'s. with absolutely continuous density. We show that for any value of the field F>0, the velocity of the t.p. converges to a limit value, which we compute.     

Rotating an interval and a circle

We compare periodic orbits of circle rotations with their counterparts for interval maps. We prove that they are conjugate via a map of modality larger by at most 2 than the modality of the interval map. The proof is based on observation of trips of inhabitants of the Green Islands in the Black Sea.

     

Well-posedness for a transport equation with nonlocal velocity

We study a one-dimensional transport equation with nonlocal velocity which was recently considered in the work of Córdoba, Córdoba and Fontelos [A. Córdoba, D. Córdoba, M.A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math. (2) 162 (3) (2005) 1377-1389]. We show that in the subcritical and critical cases the problem is globally well-posed with arbitrary initial data in Hmax{3/2−γ,0}. While in the supercritical case, the problem is locally well-posed with initial data in H3/2−γ, and is globally well-posed under a smallness assumption. Some polynomial-in-time decay estimates are also discussed. These results improve some previous results in [A. Córdoba, D. Córdoba, M.A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math. (2) 162 (3) (2005) 1377-1389].     

Boundary value problem for a second-order elliptic equation in the exterior of an ellipse

We consider a boundary value problem for a second-order linear elliptic differential equation with constant coefficients in a domain that is the exterior of an ellipse. The boundary conditions of the problem contain the values of the function itself and its normal derivative. We give a constructive solution of the problem and find the number of solvability conditions for the inhomogeneous problem as well as the number of linearly independent solutions of the homogeneous problem. We prove the boundary uniqueness theorem for the solutions of this equation.     

A potential reduction algorithm for an extended SDP problem

An extended semi-definite programming, the SDP with an additional quadratic term in the objective function, is studied. Our generalization is similar to the generalization from linear programming to quadratic programming. Optimal conditions for this new class of problems are discussed and a potential reduction algorithm for solving QSDP problems is presented. The convergence properties of this algorithm are also given.     

On a viscous Hamilton-Jacobi equation with an unbounded potential term

We present comparison, uniqueness and existence results for unbounded solutions of a viscous Hamilton-Jacobi or eikonal equation. The equation includes an unbounded potential term V(x) subject to a quadratic upper bound. The results are obtained through a tailor-made change of variables in combination with the Hopf-Cole transformation. An integral representation formula for the solution of the Cauchy problem is derived in the case where V(x)=ω²|x|²/2.     

Analytical methods for an elliptic singular perturbation problem in a circle

We consider an elliptic perturbation problem in a circle by using the analytical solution that is given by a Fourier series with coefficients in terms of modified Bessel functions. By using saddle point methods we construct asymptotic approximations with respect to a small parameter. In particular we consider approximations that hold uniformly in the boundary layer, which is located along a certain part of the boundary of the domain.     

Discrete velocity models with general boundary conditions in a slab

We consider a gas between parallel plates, described by a discrete velocity model. At the boundary, we impose the most general linear boundary conditions which preserve mass. Using a fixed‐point theorem we prove the existence of at least one parameter family of solutions, continuous in x. The velocities are assumed to have a non‐zero component in the direction orthogonal to the boundaries. Copyright © 2001 John Wiley & Sons, Ltd.     

Chiral solitons in a quantum potential

We study chiral solitons in a quantum potential using a dimensional reduction of the problem for (2+1)-dimensional anyons. We show that the integrable version of the model is described by a family of the resonant derivative nonlinear Schrödinger equations. For a quantum potential strength s > 1, we show that the chiral soliton interaction has a resonance. We investigate the semiclassical quantization procedure for solitons.     

Static and dynamic best-response potential functions for the non-linear Cournot game

We show that the Cournot oligopoly game with non-linear market demand can be reformulated as a best-response potential game where the best-response potential function is linear-quadratic in the special case where marginal cost is normalized to zero. We also propose a new approach to show that the open-loop differential game with Ramsey dynamics admits a best-response Hamiltonian potential corresponding to the sum of the best-response potential function of the static game plus the scalar product of transition functions multiplied by the fictitious costate variables. Unlike the original differential game, its best-response representation yields the map of the instantaneous best reply functions.     

A continuation and existence result for a boundary value problem on an unbounded domain arising for the electrical potential in a cylindrical double layer

Using a continuation theorem for contractions, the existence, uniqueness, and an approximation method for a class of nonlinear boundary value problems on an unbounded interval is obtained. The results apply in particular to the problem in the title.     

Mixed problem for the wave equation with a summable potential and nonzero initial velocity

The resolvent approach in the Fourier method, combined with Krylov’s ideas concerning convergence acceleration for Fourier series, is used to obtain a classical solution of a mixed problem for the wave equation with a summable potential, fixed ends, a zero initial position, and an initial velocity ψ(x), where ψ(x) is absolutely continuous, ψ'(x) ∈ L ₂[0,1], and ψ(0) = ψ(1) = 0. In the case ψ(x) ∈ L[0,1], it is shown that the series of the formal solution converges uniformly and is a weak solution of the mixed problem.     

Homoclinic solutions for nonautonomous second-order Hamiltonian systems with a coercive potential

A new existence result of homoclinic orbits is obtained for the second-order Hamiltonian systems , where F(t,x) is periodic with respect to t. This result generalizes some known results in the literature.     

Numerical simulation of the velocity distribution for a hard-sphere gas

The kinetic features of energy exchange between hard-sphere molecules are numerically simulated. The results deviate from the usual the Maxwell distribution. The high-speed tail of the molecule distribution is cut off due to the dominant deceleration of fast particles. For mixtures of gases with various molecular masses, the energy redistribution between the degrees of freedom leads to different stationary temperatures of the components.