Exact solutions are developed for instantaneous point sources subject to nonlinear diffusion and loss or gain proportional to nth power of concentration, with n > 1. The solutions for the loss give, at large times, power-law decrease to zero of slug central concentration and logarithmic increase of slug semi-width. Those for gain give concentration decreasing initially, going through a minimum, and then increasing, with blow-up to infinite concentration in finite time. Slug semi-width increases with time to a finite maximum in finite time at a blow-up. Taken in conjunction with previous studies, these new results provide an overall schema for instantaneous nonlinear diffusion point sources with nonlinear loss or gain for the total range n ≥ 0. Six distinct regimes of behaviour of slug semi-width and concentration are identified, depending on the range of n, 0 ≤ n < 1, n = 1, or n > 1. Three of them are for loss, and three for gain. The classical Barenblatt-Pattle nonlinear instantaneous point-source solutions with material concentration occupy a central place in the total schema.

For the cubic Schrödinger equation iut = uxx + k|u|2u, 0 ≤ x, t < ∞, initial data u(x, 0) = u0(x) ∈ H2[0, ∞), and Robin boundary data ux(0, t) + αu(0, t) = R(t) ∈ C2[0, ∞) (where α is real), we show that the solution u depends continuously on u0 and R.

In this article we consider modified search directions in the endgame of interior point methods for linear programming. In this stage, the normal equations determining the search directions become ill-conditioned. The modified search directions are computed by solving perturbed systems in which the systems may be solved efficiently by the preconditioned conjugate gradient solver. A variation of Cholesky factorization is presented for computing a better preconditioner when the normal equations are ill-conditioned. These ideas have been implemented successfully and the numerical results show that the algorithms enhance the performance of the preconditioned conjugate gradients-based interior point methods.

We improve some results of [17], which relate to key tools given in [7] for establishing canonical inequalities used in the analysis of sum sets and fractals.

This paper examines the predictions of shallow water theory for steady and unsteady withdrawal flows through an extended sink from fluid of finite depth. Two-dimensional plane flows and three-dimensional axi-symmetric flow through a circular drain are examined. Shallow water theory indicates the presence of limiting configurations, where the surface of the fluid collapses directly into the sink. In addition, this theory suggests that some previously computed steady solutions may be unstable.

Recently, several papers [2–4, 6] have been published concerning a pursuit problem which was apparently first posed explicitly by Leonardo da Vinci and which may have been present in earlier thinking about kinematics and geometry. Falconry appears to go back, in Europe, to the days of Pliny, Aristotle and Martial, and, in Asia, to 2000 BC [5].

We study the averaging of the Hamilton-Jacobi equation with fast variables in the viscosity solution sense in infinite dimensions. We prove that the viscosity solution of the original equation converges to the viscosity solution of the averaged equation and apply this result to the limit problem of the value function for an optimal control problem with fast variables.

The Shannon system is generalized and the expansion of a function in the generalized Shannon system is considered. No study of a wavelet expansion exists without the assumption of ‘fast’ decay of wavelets. The wavelet ψ which is associated with the generalized Shannon system has a ‘slow’ decay. The expansion of a function in the system is shown to converge at a point which satisfies the Lipschitz condition of order α > 0. On the other hand, there is a continuous function whose wavelet expansion in the generalized Shannon system diverges. An observation of Gibbs' phenomenon is also given.

An inequality involving the logarithmic mean is established. Specifically, we show that

where . Then several generalizations are given.

Simple chemical reactions can be described by the Michaelis-Menten response curve relating the velocity V of the reaction and the concentration [S] of the substrate S. To handle more complicated reactions without introducing general polynomial response curves, the rate constants can be considered to be scale dependent. This leads to a new response curve with characteristic sigmoidal shape. But not all sigmoidal curves can be accurately fit with three parameters. In order to get an accurate fit, the lower part of the ∫ shaped curve cannot be too shallow and the upper part can't be too steep. This paper determines an exact mathematical expression for the steepness and shallowness allowed.